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AS Level Mathematics 9709 | Coordinate Geometry 7 | Standard Equation of Circle | Raja Shafqat — Transcript

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0:08Assalam-o-Alaikum, son. From today, we
0:10are finally going to start the second
0:12part of Coordinate Geometry. The
0:14Coordinate Geometry we studied
0:16previously is what we call simple
0:18Coordinate Geometry. Is that clear, son
0:21? The Coordinate Geometry we studied
0:23before this is what we call basic
0:24Coordinate Geometry. Alright? The
0:27Coordinate Geometry we are studying
0:29today is called advanced Coordinate
0:30Geometry. For the Coordinate Geometry
0:32from today onwards, we use the
0:34mathematical term advanced Coordinate
0:36Geometry. Is that clear, son? In
0:38mathematical language, this is called
0:40Coordinate Geometry of Circles. Is that
0:42clear, son? So, the Coordinate Geometry
0:46we are finally starting today is named
0:49Coordinate Geometry of Circles. Alright
0:53, son? This is called Coordinate
0:55Geometry of Circles. In the first class
0:58, during the introduction, I explained
0:59that there are two types of Coordinate
1:01Geometry in O-levels. One is read as
1:03basic Coordinate Geometry and the other
1:05is read as Coordinate Geometry of
1:07Circles. So today, our main target is
1:09essentially the second point of
1:12Coordinate Geometry, which is called
1:14Coordinate Geometry of Circles. Now,
1:17first of all, what are the requirements
1:19for this topic? Let's take a look at
1:20that first. Then I will guide you
1:22slowly. So, the topic we are starting
1:26today. Well, the first thing is, when
1:27was this added to the course? It was
1:29added to the course in May/June 2020.
1:32Is that clear, son? And ever since then
1:34, basic Coordinate Geometry does not
1:36appear in the exam anymore. Now, basic
1:39Coordinate Geometry does not come. Now,
1:40Coordinate Geometry of Circles comes.
1:43Because in Coordinate Geometry of
1:45Circles, the entire basic Coordinate
1:47Geometry is also covered. Is that clear
1:50, son? The entire basic Coordinate
1:53Geometry is covered in this. Alright,
1:55son, the entire basic part. Now look,
1:57what are the requirements for this?
1:58Listen to this first. The things
2:00required for this are what we have
2:01already studied. I will guide you one
2:03by one. You need to know two or three
2:05things for this. Number one, the first
2:08requirement for this topic is basic
2:11Coordinate Geometry. Alright? We
2:13studied basic Coordinate Geometry
2:15previously up until the last class. If
2:16we remember, what did we study in basic
2:18Coordinate Geometry? In basic
2:20Coordinate Geometry, we studied the
2:23distance formula, length of a line
2:25segment and ratio, finding an unknown
2:27point using ratio, midpoint, gradient,
2:30equation of a straight line, equation
2:32of a perpendicular bisector, and
2:34Heron's formula. These seven points are
2:36what we call basic Coordinate Geometry.
2:38So, basic Coordinate Geometry is
2:39required for this. Number one. Then,
2:41geometrical properties of circles are
2:43required for this. Alright, son? So,
2:45what is required for this? Geometrical
2:48properties of a circle. We studied a
2:50topic on geometrical properties. The
2:53chapter's name was Circular Measure. So
2:55, circle properties are also required
2:58for this. Then, one more thing is
3:00required for this topic. When we
3:01studied chapter one of the course, its
3:03name was Quadratics. So, some
3:05quadratics will also be used here. Okay
3:08? If we explain in more detail what
3:10quadratics will be used? Then you will
3:14use number one, let's say, completing
3:16the square. Let's start from here.
3:21Number one, you will first find
3:23completing the square in this.
3:25Completing the square. This will also
3:28be used in this: completing the square.
3:30Then what else will you find in this?
3:32You will find the relationship between
3:35a line and a curve in this.
3:38Relationship between a line and a curve
3:42. This point about quadratics is a must
3:44. The relationship between a line and a
3:46curve will definitely be used here.
3:48Then you will find a third point in
3:50this. That point is called quadratic
3:53inequalities. Quadratic inequalities.
3:56Is that clear, son? So, this is
3:58basically a game of these points. Basic
4:00coordinate geometry is 100%required.
4:03Circle properties are required.
4:05Completing the square from quadratics
4:08is required. Relationship between a
4:09line and a curve is required. And we
4:11studied one more topic. The topic's
4:13name was quadratic inequalities. Clear?
4:17It is a must that before starting this
4:19topic, you should know all this and
4:22have already done it. Okay, son? Now, I
4:25will finally have you start the topic.
4:28There are basic five points in this
4:30topic. By the way. There are basic five
4:33points. And after that, all these
4:35things will be mixed up in those five
4:37points. Clear, son? Meaning, basically
4:40in this entire topic, you will find
4:42five cases or five points. Plus,
4:44everything will be repeated or combined
4:47with those five points. Come, now let's
4:50start even slower. Well, the very first
4:53thing, the first point of this topic
4:55that we are starting today, is called
4:57the standard equation of a circle. I
5:00will finally start this topic from here
5:02today. So, the first point of this
5:04chapter is called the standard equation
5:06of a circle. Standard equation of a
5:09circle. Let's go. By the way, you might
5:11not have heard the term "equation of a
5:13circle" until today. You must have
5:15heard the term "equation of a line" or
5:17the equation of any curve. What is
5:18required for us today? Today we require
5:20the equation of a circle. So, I will
5:22start with a small point. Then I will
5:24guide you forward one by one. Our
5:26interest today is in the equation of a
5:29circle. So, I'll show you a small
5:30derivation. It's not required. But
5:32we'll check it out. Suppose this is
5:35your circle and this is the center of
5:36the circle. Like we normally mark.
5:39Suppose you have the center of the
5:41circle, and the gap from the center of
5:43the circle to the boundary is obviously
5:45what we call the radius. Obviously, we
5:47call that the radius. Here it is. So I
5:49considered the coordinates of the
5:52center as (a, b). Let me explain it to
5:54you this way: I assumed the coordinates
5:56of the circle as (a, b); what are the
5:58circle's coordinates?(a, b)–obviously
6:02, what is the center of the circle? The
6:04center of the circle is basically a
6:06point. The center of the circle is
6:09basically a point. Okay, then I
6:11considered another point on the
6:12circumference. Here you go. I
6:14considered another point on the
6:16circumference. Right? Suppose this is
6:18your center of the circle. And I
6:20considered a point P on the
6:22circumference. So let me mark here, by
6:25the way, what is C? To explain this to
6:28you, I will also copy these basic
6:30points: what is C? C is, suppose, the
6:34center of the circle. C is, suppose,
6:37the center of the circle. Right? What
6:39is C? It is a point. So what is (a, b)?
6:42Coordinates of the center. Coordinates
6:45of the center. Right? So what is this?
6:47What is C? C is the center of the
6:49circle. And what are a and b? What do
6:51these a and b show? Obviously, who is
6:53the center of the circle? The center of
6:56the circle is obviously C. And what are
6:58a and b? a and b are the coordinates of
7:01the center. a and b are the coordinates
7:06of the center. Meaning, if the center
7:09of the circle is a point, it will
7:11obviously have both x and y coordinates
7:13. Okay. And I considered a point, any
7:16point on the circumference. I named
7:18this point P. The name of this point is
7:20point P. I considered a point. Any
7:23point on the circumference. So I named
7:25that point (x, y). Here you go, I just
7:28randomly took a point, a point on the
7:30circumference, and named that point (x,
7:32y). So if we use a little common sense,
7:36the gap between this point C and P, the
7:39distance, will indirectly show what? It
7:42will show the radius. Right? What will
7:45it indirectly show? The radius. So,
7:48this means I can guide you that the
7:49distance between P and C, what will
7:51that distance indirectly show? That
7:54distance will indirectly show the
7:56radius. That distance will indirectly
7:58show the radius. Even if you use common
8:01sense, this point here is a, b. This
8:04point is your x and y, so this gap is
8:06what we call the x-difference, and this
8:08gap is what we call the y-difference.
8:11Clear? Basically, this is a line, this
8:13is a line segment, by the way, here it
8:15is. Let’s say this is the center of
8:17the circle and this is any point on the
8:18circumference. So, if I apply the
8:20distance formula to this. Come on, we
8:22must have studied the distance formula
8:24in coordinate geometry. We must have
8:26studied the distance formula in
8:28coordinate geometry. We must have
8:29studied it in O-Level Math and
8:30Additional Math. (x2-x1) ² + (y2-y1)
8:33², so I can also show it like this: if
8:35I ask you, "Hey, I need the distance
8:37between P and C." Then what will you
8:40apply? You will apply the distance
8:42formula. Here you go. And what will the
8:44distance formula be equal to? Radius.
8:46So, what is the distance formula? The
8:49distance formula will be square root of
8:53(x-a) ² + (y-b) ². What have we
8:55applied here? The distance formula. Now
8:58, if I make it a bit easier, just a bit
9:01easier, I applied squaring on both
9:03sides. Squaring both sides, which means
9:07the square root got canceled out. So
9:09what is this? This is (x-a) ². Plus,
9:14what is this? This is (y-b) ². (x-a)
9:20² + (y-b) ² = radius squared. From
9:25today onwards, in the language of math,
9:28we will read this equation. What is
9:30this? This is the standard equation of
9:33a circle. We read this equation in the
9:35language of math. Standard equation of
9:38a circle. Clear, kid? Now, look at its
9:40language carefully. We read this
9:42equation in the language of math.
9:45Standard equation of a circle. So, by
9:47the way, we don't need its derivation.
9:49But the derivation was very basic. So,
9:52I will explain it again. What was x and
9:54y? x and y was any point on the
9:56circumference. Point on the
9:58circumference. And what were A and B in
10:01this? A and B were the coordinates of
10:03the center. A and B are the coordinates
10:05of the center. Alright? A and B are the
10:07coordinates of the center. And what was
10:09R in this? R for radius. R for radius.
10:13So, take another look at it. Let me
10:15repeat this once again. What are x and
10:17y? x and y is any point on the
10:19circumference. What are a and b? a and
10:23b are the coordinates of the center.
10:25Coordinates of the center. What is a, b
10:28, the coordinates of a, b? It is a
10:30point, by the way. And what is C
10:32representing here? C represents the
10:33coordinates of the center. What is C
10:35representing? Coordinates of the center
10:38. Okay? And what is R representing? R
10:40represents the radius. R for radius.
10:44Now, first of all, look at one more
10:45thing. In this equation, the standard
10:48sign is negative. Here too, the
10:50standard sign in the center is negative
10:53. And what is the sign between these
10:54two whole squares? Positive. Take a
10:57good look at this once. Inside the
11:00bracket, what is the coefficient of x?
11:02One. Inside the bracket. What is the
11:05coefficient of y? One. Note this.
11:07Observe this carefully one more time.
11:09What is the sign in the center? The
11:11sign in the center is plus. And what is
11:14this sign in the middle? Minus x-a²
11:18y-b² = r², so what is this? This was
11:23the x difference. What is this? This
11:25was the y difference. And this formula
11:27is basically the Pythagorean theorem.
11:29It is the Pythagorean theorem. Clear,
11:32son? It is the Pythagorean theorem. If
11:34you use a little common sense, just a
11:36little. Then you see that basically,
11:39what are C and P forming? This C and P
11:41are forming a right-angled triangle.
11:43This was your point. This was your
11:46point X Y, and this point C, its
11:48coordinates are A B, so obviously, what
11:51is the horizontal gap between C and P?
11:54The horizontal gap is X-A. Common sense
11:58. And note one more thing. What is the
12:01vertical gap between C and P? y-b y-b,
12:05here you go, y-b, so that's why the
12:09radius was, by the way, the radius was
12:12your hypotenuse, so what was derived is
12:16x-a² plus y-b² equals r², so please
12:20keep in mind how we read this equation.
12:24The standard equation of a circle, what
12:26is the coefficient of x and y inside
12:28the bracket? The coefficient of x and y
12:31is always one. What is the sign in the
12:33center?-x-a² y-b² and what is this?
12:38What is between these two whole squares
12:40? There is a plus sign between these
12:41two whole squares. And what is r? r is
12:44the radius. r is the radius. In the
12:47language of math, we read this equation
12:49as: Standard equation of a circle.
12:51Alright? This is our first point from
12:53which we are starting our coordinate
12:55geometry. Coordinate geometry of
12:57circles. Clear? Okay, now come, let me
13:00finally show you some examples of this
13:02first. First some basic questions so
13:03that we get a bit used to it. Then, God
13:05willing, I will start further. Look son
13:08, I will copy it again. So, what did I
13:11tell you today? Today we studied a
13:14small new point. Just like we normally
13:16study the equation of a line in math.
13:18So today we learned a new thing: the
13:20standard equation of a circle. So what
13:21did I explain to you regarding what the
13:23standard equation of a circle is? I
13:25will copy it again once. It's always
13:27x-a² y-b² = radius squared. So I
13:36deliberately copied this equation again
13:38so that it repeats a little bit. We
13:41should get some idea of how it will be
13:43used. Clear, son? Come on, let's
13:46finally start. First, some very basic
13:49one-mark questions. Find the center and
13:52the radius. We know what a and b are,
13:55right? Center, what is r? Radius, and
13:58what are x and y? Any point on the
14:01circumference. So now you see, let's
14:04start. Basically, what do you have to
14:06do now? You have to compare all these
14:08questions. What is your task for all
14:12these given questions? What is your job
14:14? Whatever given questions these are,
14:16what do you have to do with all of them
14:18? Compare with the standard form.
14:22Compare with the standard form so that
14:24you can find their center and radius.
14:27Clear, son? Be able to find the center
14:29and radius. Now let's start with the
14:31first example. Come on, now look at
14:33part A of this question. In part A, you
14:35are already given a circle. What is the
14:38circle? x² + y² = 16, here it is.
14:43What do you always have to compare it
14:45with? You always have to compare it
14:46with only one thing. And the name of
14:48that one thing is the standard equation
14:50of a circle. When you compare it with
14:52this, your target is that you always
14:54have to compare it with this. Here it
14:56is, you have to compare with this. Here
14:58it is. So now look at your given
15:00equation. When you check your given
15:02equation, here it is. If you have even
15:05a little common sense, any child can
15:07say that I can express this given
15:08equation like this. x-0² y-0² x-0²
15:15here it is. y-0² = 4², you can
15:20express it. It is the same thing. What
15:23is x-0? What is y-0? What is the square
15:28of 4 in y²? 16, so now when you match
15:31it, what is your a? 0, if you match it,
15:34what is b? 0, so from here I found out,
15:37what are the coordinates of the center?
15:390, this is your coordinates of the
15:41center. And even if you use common
15:44sense, what is the square of the radius
15:45? The square of the radius is 16, so
15:48then what is the radius? The radius is
15:50four. Think for a minute. If the square
15:53of the radius is 16, then obviously
15:55what will the radius be? Obviously, the
15:57radius will be four. So, this was a
15:59very basic question. Very basic. So,
16:03this means if any student uses common
16:05sense. Even if you use common sense. If
16:08in your question, the center of this
16:11equation's circle becomes 0. What does
16:14this indirectly mean, that your circle
16:16is such that its center is? 0 0,
16:18whatever the radius may be, what is the
16:21center? It became 0. So, if the
16:24circle's center, obviously the center
16:25is a point. The point can be anything.
16:29If we have common sense, the center can
16:31be here, here, here, or here; the
16:33center is basically a point. And in
16:35math, there are nine points. By the way
16:37. If we have sense, some points are
16:40between the axes. Some points are on
16:44the axes. Now, whether it is on x or y,
16:47and there is one special point, the
16:49origin. So, if in your question the
16:52center's coordinates become 0, then
16:54common sense says what will your
16:57equation always be? Common sense says
16:59that then what will your equation
17:01always be? x² + y² = r² The center
17:06can be anything. I am not saying it
17:08will always be the origin. The center
17:10of the circle can be anything. I am not
17:11saying it will always be the origin.
17:13But if it is the origin, then this and
17:15this become what? They become zero.
17:16What are a and b? Coordinates of the
17:18center. So, what is left? x² + y² = r
17:22squared. So, basically what do you have
17:25to do? Basically, you have to compare.
17:27Clear, beta? Just compare. Look at the
17:31second part of this. The second part of
17:33this question. By the way. Right now,
17:35we are just in the introductory phase
17:37where equations are already given. We
17:39are just comparing. By the way. Just
17:41comparing. Now, you see one more
17:44equation given here, 2 x² + 2 y² = 9,
17:47here it is. You note one thing. There
17:50should always be a one with x². Always
17:52one with y². Meaning the whole square
17:55and this whole square, the coefficient
17:58with both must always be one. Please
18:01take a good look at the standard form
18:03once. Inside, there should be a one
18:05with x, and the coefficient of the
18:06whole square must also be one. Inside
18:08the bracket, there should be a one with
18:10y, and it should be one with the whole
18:13square too. Clear, kid? So now, a
18:15little extra common sense will have to
18:17be used here. Take two as a common
18:19factor. And shift the two there to do
18:21what? Divide it. Here you go. Shifted
18:24the two and divided. Now this means
18:26that what are your a and b? Your A is
18:29also zero, your B is also zero. Let me
18:32make it even easier. I will force it to
18:34be shown like this. I am showing it
18:36this way on purpose. What does common
18:38sense say x² means? Any child can show
18:40x² like this too. x-0 whole squared,
18:43here you go. And any child can show it
18:46like this as well. Any child can show
18:48this y-coordinate in the same way. How
18:50would you show the y-coordinate? You
18:52would say y-0 whole squared. Here you
18:54go. The same thing again. I have
18:57mentioned two or three times above that
19:00if the circle's center is 0, then x² +
19:03y² = r² will be your circle. Equal to
19:06what did it become? It became 9/2. Here
19:08you go. Now when any child compares
19:11this with the standard equation. Look,
19:13the name of this equation is standard
19:16equation. So far in the lecture, we
19:18have only talked about one equation.
19:21And the name of this equation is
19:22standard equation. Now when you match
19:24them, what is a? Zero. What is b? 0, so
19:26it is confirmed that what are the
19:27coordinates of the center? 0, here you
19:29go, so what is the center of your
19:31circle? 0. Now tell me one thing,
19:36radius squared is 9/2, obviously what
19:38is the radius? Radius is a length. So
19:42when I apply the square root, obviously
19:44when I apply the square root, length is
19:46always positive. But the answer will
19:49come out both plus and minus. Plus as
19:51well as minus. Alright, so when I apply
19:53the root, it becomes; what is the root
19:55of nine? Three, and the root of two
19:57will be root 2, but obviously, length
20:00is always positive. So when I talk
20:02about the radius, I will always, for
20:03the negative sign of the length, for
20:05the negative sign of the radius, I will
20:06always do what? I will ignore it. So,
20:08what did we find out here? We found the
20:10coordinates of the center and the
20:12radius. These were very basic questions
20:14. In both these questions, you had to
20:17identify the circle's center and its
20:19radius. The standard equation is (x-a)
20:23² + (y-b) ² = r². I repeated it four
20:26or five times that if a and b become 0,
20:30the equation becomes this. There it is,
20:33the equation of the circle becomes this
20:35. Now, look at part C of this same
20:37question. We are currently in round one
20:40. We are at the basic introduction. So
20:42that a sense is developed once. What is
20:44a? What is b? What is the center? What
20:47is x? What is y? Inside the bracket,
20:49there should be a one with x and y.
20:51Outside the bracket, for (x-a) ² and (
20:54y-b) ², there should be a one with
20:56both. Now look at another example of
21:00this question: x² + (y-2) ² = 25,
21:05just a little common sense. We know
21:08that our form involves a whole square,
21:10right? So I can obviously show it like
21:13this. If you note, I can also say it
21:15like this. (x-0) There it is, (x-0) ².
21:19Obviously, this means x², and what
21:22does this mean? This means (y-2) whole
21:26square; it is already in the whole
21:28square form. What does this mean?(y-2)
21:30² This is already in the whole square
21:34form. Equals 25, right? Equals 25,
21:37right? It's not rocket science. It is
21:39nothing extraordinary. Now, when a
21:41student compares this with the standard
21:43form. This is your standard form. We
21:45have only studied one equation so far.
21:47The standard form is right here. Now,
21:49when the student matches them, what
21:50does a become? 0 will match. What does
21:53b become? Two. So, from here, I found
21:55out the center coordinates are? 0, 2.
21:57There you go. From here, I found out
21:59what the center coordinates are. 0, 2.
22:01Tell me, what is the radius squared?
22:05The radius squared is 25. Here, the
22:09radius squared is 25. There you go. So
22:11if the radius squared is 25, the radius
22:13would be plus or minus 5, but obviously
22:15, the radius is a length. So length is
22:18always positive. Length is always
22:20positive. So why are we doing this
22:22question now? To develop a sense within
22:26us to identify how to find the circle's
22:29center and radius by comparing? Because
22:33this will definitely be used later. Now
22:35, look at part D. In part D, the whole
22:37square form is also already given to
22:39you. In part D as well, the whole
22:41square form is already given to you.
22:43Here is the whole square form for you.
22:45A new issue this time. If you compare
22:48this with the standard form, look, (x-a
22:51) squared. We studied the same thing in
22:53quadratics as well. The topic name was
22:55curve sketching. What did we use to do
22:57in that too? We used to compare it with
23:00the standard form. Look here: x x, done
23:04, minus minus, done. So, it is
23:07confirmed from here that what is the
23:09answer for a? a is 5. Simply match y
23:13and y. Or let's do it this way. If a
23:15student's math is very weak, they will
23:18just cancel it out. They will say-b = 3
23:21, so what is the answer for b from here
23:23? The answer for b came out to be 3.
23:26The answer for b came out to be sorry,
23:28-3. Or you can write it all out again
23:30by comparing it like this. In place of
23:323, we have-b. So then tell me, what is
23:34b? Your b is -3. Your b is -3, so what
23:37are the coordinates of the center? The
23:40coordinates of the center are (a, b),
23:42so what did your a become? Your a
23:44became 5 and what did your b become?
23:46Your b was found to be -3. Clear, beta?
23:49Both a and b were found, meaning the
23:51center coordinates were found. So now
23:53tell me, what is the square of the
23:54radius? The square of the radius is
23:56four. So when you apply the square root
23:58, obviously it will always be
23:59plus-minus. But the radius is a length.
24:02So length is always positive. Length is
24:05always positive. So these four
24:06questions were very basic questions.
24:08Very basic indeed. The question is, why
24:11did we do them? So that a sense
24:13develops within us on how to identify
24:15the center and radius if an equation is
24:18given. I will just repeat one thing.
24:21What is the sign in between? x and y
24:23minus, (x-a) squared. (y-b) squared, x
24:27squared coefficient 1, whole square
24:30coefficient 1. Inside the bracket, the
24:34coefficient of x is 1; inside the
24:37bracket, the coefficient of y is always
24:401. So I am repeating things again and
24:43again so that you remember what we did.
24:45Clear, beta? So what did we do simply?
24:47We were already given four equations.
24:49We compared them with the standard
24:51equation. Just matched them, compared
24:53them. We made no extra effort. What was
24:55our only task? To compare. Clear, beta?
24:57Come, let me show you a slightly better
24:59question. Here you go. Then I will take
25:03you one by one to slightly tougher
25:05questions. What does tougher questions
25:07mean? When you have to apply completing
25:09the square. I will explain what
25:10completing the square is in a little
25:12while. Okay? So, let me guide you once
25:15more on what you have studied today. So
25:17far, you have learned with me that we
25:21use an equation for a circle. We call
25:24this equation the standard equation of
25:27the circle. And that is all you have
25:30studied with me so far. What do we call
25:32this equation in the language of math?
25:35Standard equation of a circle. Other
25:37than that, we haven't studied anything
25:40else yet. Is that clear, kid? We are
25:43still at a very basic point, and I am
25:45copying the equation repeatedly so that
25:48we get used to it. So that we develop a
25:50sense that yes, this is the standard
25:52equation. Then we will see how to apply
25:54it in different situations. Now, look
25:57at part E of this same question. When
26:00you check part E of this question, what
26:03is part E? x + 7² y² = 18, here it is
26:09. Now, what do you have to compare it
26:11with? You always have to compare it
26:13with the standard form. Look, we know
26:15that we always compare it with this.
26:19x-a² + y-b² = radius squared. Is that
26:25clear, kid? Now, compare it and tell me
26:27what-a is. When you compare it, you
26:31will say-a is 7. So then, what does a
26:33become? a becomes -7 because there must
26:36always be a minus sign in the center.
26:38So indirectly, you can also say x-(-7),
26:40so what is your a? Your a, by the way,
26:44is -7; we found a. And besides, b is
26:47very easy. This y² that you have, you
26:49can also write it indirectly like this.
26:52y-0², is that clear, kid? So, it is
26:55confirmed from here that what is your B
26:57? B is zero. B is obviously so simple
27:00that there is no need to do any work
27:02for it. It is already obvious what its
27:04B is. Its B is zero. Here it is, its B
27:07is found. B has also been found. Now
27:09tell me, what is the radius squared?
27:11From this, I found out that the radius
27:13squared is 18. So when the radius
27:16squared is 18, what is the answer for
27:18the radius? 18. There you go. The
27:21radius squared is 18. So what is the
27:23answer for the radius? The answer for
27:25the radius is root 18, either express
27:27it in this form or if you want to make
27:29it easier, simplify it further. What is
27:3118? 9*2 = 18 9*2 = 18, so let's make
27:34this a bit easier. The square root of 9
27:37is 3. So, I have represented it in this
27:39surd form. So, what did its radius
27:41become? 3√2, so if I summarize, what
27:44is the center of this circle? -7, 0,
27:47and if I summarize, what is its radius?
27:513√2, so notice one thing, what are we
27:53doing in all these questions? We
27:56compare our given question with the
28:00standard form and just identify what 'a
28:04' is. What 'b' is. And what the radius
28:08is. Clear, son? We have no interest in
28:11x, y. x, y represents any point on the
28:13circle. On the circle. Our interest is
28:15in the center of the circle, meaning 'a
28:18', 'b' and the radius; these two things
28:20are what we are interested in. Now look
28:23at part 'f' of this same question. Okay
28:27? Look at part 'f' of this same
28:28question. Now, this one is a bit
28:30interesting. It is written here as 2
28:33times (x-3) ² + 2 times (y + 4) ²
28:37equals what? 45. We already know that
28:40there should be a one with the whole
28:42square. But there is a two here. There
28:45should be a one with the whole square.
28:47But there is a two here. So, what is
28:49the first thing any student will do?
28:52The first thing is you took two as
28:53common. There you go, what you did was
28:56from your bracket, which is your left
28:58side, you did what with the two? You
29:01took two as common from it. When you
29:03took two as common, what did your left
29:05side become? By the way. When we took
29:07two as common, our left side became (x-
29:123) ² + (y + 4) ². Bracket closed,
29:17equals what? 45. There is no
29:19extraordinary work here. We have only
29:20and only done one thing. We have only
29:22and only taken two as common. Because
29:24we know that the coefficient of the
29:25whole square should be one. Now, shift
29:27the two there and divide. When you
29:30shift the two there and divide, it
29:32becomes (x-3) squared. It becomes (y +
29:344) squared. = 45/2. The question has
29:38converted into this shape. Now when you
29:41compare it with this form, this form is
29:43called the standard form. You have to
29:47compare it with this. When you compare
29:50it, you will say x is x, minus is minus
29:51. So, from here I confirmed what 'a' is
29:55. Your 'a' is three. There you go, it
29:57is compared. Next, y is y. So I found
30:01out that-b = 4. So tell me, what is b?
30:04B is -4. There is no need for so much
30:06effort. Still, if you want me to show
30:08every step, then let me show it like
30:11this, that this means-b = what? It
30:13became four. So what is the derived
30:15answer for B? The derived answer for B
30:17is -4. What is the derived answer for B
30:19? -4. Look here, what is the radius
30:22squared? From here I found out, what is
30:24the radius squared? The radius squared
30:27is 45/2. There you go. So then, what
30:30will the radius itself be? The radius
30:32will be its square root. There you go.
30:34So what you do is the square root,
30:35meaning 45, by the way, what is it? 45
30:37is 9*5, and by the way, if you put it
30:40in the calculator, the calculator
30:42itself will show the answer in exact
30:44form or decimal. The root of 9, by the
30:47way, is three. So if 3√5/2, then what
30:50was found out from here? The radius of
30:54the circle was found out from here, so
30:56it is nothing extraordinary. In all
30:59these questions, what did we do? We
31:01just compared the given question with
31:04the standard form. Just compared it.
31:07Now tell me one thing, what are the
31:08coordinates of the center? a b, here it
31:10is. And what did its radius come out to
31:12be? Its radius came out to be 3√5/2,
31:16so both your center and radius were
31:20found. The center was found and the
31:24radius was found. The center was found
31:28and the radius was found. So, what were
31:31we technically doing until now? Until
31:34now, technically, whatever question was
31:36given to us, we were just comparing it
31:38with the standard form. Now I am
31:41telling you something new. It will be a
31:43little bit tough. Clear, beta? It will
31:45be a little bit tough. Come on. Now the
31:48situation here is not ideal. Clear,
31:51beta? Now you will have to put in extra
31:53effort here. Okay, beta? We know, I
31:56have already told you what the standard
31:58form is. Should I explain it again? Now
32:01here, for the first time, you will have
32:03to put in some extra effort. We know
32:05when we talk about the standard form,
32:07the standard form is (x-a) ² (y-b) ²
32:12= radius squared. In the previous six
32:17examples, we have done this same work.
32:22In the previous six examples, we did
32:24the same thing. Clear, beta? In all the
32:26previous examples, we did exactly this.
32:29Now there is something new. Now it is
32:32required because the issue is that the
32:34question is not given in the whole
32:37square form. It is not given in this
32:40form. Is that clear, son? The question
32:43has been expanded. So now, what is
32:45required for these two questions? Now
32:48you will have to use a little common
32:50sense. You will find this same thing in
32:52past papers as well. So in both these
32:56parts, G and H, you will have to use
32:59completing the square. And by the way,
33:03we have already studied this. The topic
33:04name, the chapter name was Quadratics.
33:06So we have to use the completing the
33:08square method. Completing the square
33:11method. Now your Quadratics will be
33:14repeated. By the way. If there is a
33:16student who knows how to compute, there
33:18used to be four rules for completing
33:20the square. If we remember, I will
33:23quickly recap the rules for you. Rule
33:25number one we always studied was that
33:28the coefficient of x² must always be
33:31+1. If it is not like that, then what
33:33happens otherwise? It will be common.
33:35We studied that the coefficient of x²
33:38must always be one. Otherwise, we
33:40studied that what would you have to do
33:42then? You have to take a common factor,
33:45and it will be possible. I am talking
33:47about completing the square. Okay? I am
33:50talking about completing the square.
33:52Clear, son? We have already studied
33:53what completing the square is. Okay,
33:55son? We have already studied it. Next,
33:57we studied another rule about how you
34:00used to find b. The formula for b was
34:04the coefficient of x divided by 2;
34:06whatever the coefficient of x was, we
34:09used to divide it by 2. Clear, son?
34:13Whatever the coefficient of x was, we
34:15used to divide that coefficient by two.
34:18We have studied this too. By the way.
34:20This is also not something new. We used
34:22to divide the coefficient of x by two.
34:25I am talking to you about old things.
34:27So that you can remember completing the
34:29square once. By the way, we practiced
34:31completing the square a lot. Still, I
34:33will revise everything once again. Then
34:36God willing, I will show you its
34:38examples. What used to be the third
34:40rule of completing the square? The
34:42third rule was what we used to do in
34:43completing the square. Add and subtract
34:46b²; we used to find the square of b
34:49first. Then we used to add and subtract
34:52its square. Then what was the last rule
34:54? The last rule was this. Then we used
34:56to apply another fourth rule. What was
34:58the fourth rule? That if a constant was
35:01involved in the question, like 2, 3, 4,
35:055, or C, we always used to copy the
35:08constant as it is in completing the
35:11square. These four rules we had studied
35:15. Where did we study them? The topic
35:16name was Completing the Square. So what
35:18is required for these two questions
35:20today? Completing the square is
35:22required for us. Is it clear, kid?
35:24Completing the square is required for
35:26us today. Now let's see one by one.
35:30Let's start with part G first. Is that
35:32okay, kid? This type of question comes
35:35for three to four marks in your past
35:38papers. They come for three to four
35:41marks in your past papers. Now let's
35:43start. Now you see x² + y²-8x + 20y +
35:49110 equals what is given? Equal to 0 is
35:55given. Clear, kid? This hint is already
35:57given to you. Done. Alright, now the
35:59first task is that in the whole square,
36:02x comes first. Then in the whole square
36:04, y comes. Then here is the square of
36:06the radius. Is it clear, kid? Then what
36:08is the radius? It is squared. Now, the
36:11first task is step number one. Step
36:13number one is this. Step number one is
36:16that what will you do first? You will
36:19copy the x terms. You will combine them
36:22. Meaning, first you separated both x
36:24terms. Then what will you do? Then you
36:28will combine or collect the y terms. I
36:31collected the y terms and copied the
36:34110, i.e., the constant, as it is. An
36:38interesting thing is that whenever you
36:40express it in this form, you will
36:42always have to apply double completing
36:45the square. Meaning you will have to do
36:48completing the square for x and also
36:50for y. Because there is a whole square
36:53on x and what is on y? Whole square. So
36:55that means you will have to use the
36:57double completing the square form. Now
37:00let's start slowly. If we know that in
37:02completing the square form, you have to
37:04convert the expanded form back into a
37:06whole square. You have to convert the
37:09expanded form back into a whole square.
37:11Now let's see, kid, what we will do.
37:13First thing, what is this? a² is in
37:15the center. Look, if the formula here
37:18is minus, it means this formula will
37:20also convert into (a-b) ². What used
37:22to be in the center? 2*a*b, there you
37:24go. Now, the center in the formula was
37:28always 2*a*b, so if we remember the '
37:30completing the square' form, we used to
37:33divide the coefficient of x by two to
37:35find b. To find b, this becomes four,
37:38so 4 squared will be added and 4
37:40squared will also be subtracted. Here
37:43you go, 4 squared will be added, 4
37:46squared will be subtracted. Is that
37:48clear, son? So we added b² and also
37:51subtracted b². Next, next. Now, you
37:57will do the same working for y as well.
37:59You just did this working for x, right?
38:01Now you will do it for y as well. Look
38:03here, a², if you note the y terms as
38:06well. Look, I will show you the same
38:09work for y as well. Here it is. If you
38:11check for y too. Here is y², so I can
38:13show it like this. Here is y². If you
38:17have a plus here, then it obviously
38:19means the formula will also have a plus
38:20. So what used to be in the center for
38:24completing the square? 2*a*b, so that
38:27means what will b be? It was 2*a*b,
38:30right? So what is your a? Your a is y.
38:32So what will b be? How will b come?
38:36Coefficient of y/2, here it is, it is
38:3810, so from here I found out that its b
38:40is 10, then what did we do? We added b
38:43squared and subtracted it, and 110 is
38:46copied as is, here it is. 110 is copied
38:49as is, here it is. Now tell me, what
38:52did this whole expression become? This
38:54whole expression converted into a whole
38:57square. Clear? This whole expression
38:59converted into a whole square. Here it
39:01is, this whole expression converted
39:04into a whole square. Similarly, this
39:06whole expression also converted into a
39:08whole square. Here you go. We know,
39:10right? The formula is complete. This is
39:12the work that needs to be done in this
39:13method. What is the name of the method?
39:15The name of the method is completing
39:17the square. Clear, son? The name of the
39:19method is completing the square. So now
39:21tell me, what did this whole thing
39:22become? What did we used to call this?
39:24I can copy this now, that what did this
39:26whole thing become? This whole thing
39:28became (a-b) ². Here you go, this
39:32whole thing became (a-b) ², meaning
39:34this entire expanded form, we packed it
39:37back up. What will be the square of -4?
39:40-16, done. Now, this entire expanded
39:44form, a²-a² + 2ab + b², if I
39:46remember the completing the square form
39:49, what will it convert to?(y + 10)
39:53squared, there you go. And what is this
39:57thing? -100 copied as is. What is this?
39:59What is the square of 10? 100 and + 110
40:01copied as is, which we also copied the
40:04same way above. Now make this a little
40:07easier. When we make this a little
40:08easier. When we make this a little
40:11easier. Then what will it be here? This
40:15will become (x-4) ², there it is. Here
40:17it will be (y + 10) ². Here it is, and
40:21this will be 116 + 110, meaning this
40:24will be -6. Equal to what? 0? There you
40:28go, -6 equal to what? 0. Clear, kid?
40:32Make this even easier. When we make
40:34this even easier, it becomes (x-4) ² (
40:38y + 10) ² = 6, there you go, your
40:42question is now converted to standard
40:44form. Now what is your task? Now your
40:47task is to match it with this. Look
40:49here, x-a becomes what? 4, so now when
40:53we compare this, we will say what is a?
40:56a is 4. Now when we match this, what is
41:00y-b equal to? y + 10, so what is-b
41:04equal to?-b is equal to 10. So it's
41:07common sense, there is no need to even
41:09show this step. So from here I found
41:11out what b is? b is -10. From here I
41:14found out what b is? b is -10, so a is
41:17also found. b is also found. Now tell
41:20me what is r squared? r squared is six.
41:26From here I found out what r squared is
41:28? r squared is six. So we know that
41:30radius is always positive. Radius
41:32refers to length. So from here I found
41:34out what its radius is? Its radius is 6
41:37. So what was your target by the way?
41:40By the way, the target was center and
41:42radius. So your radius is found. Here
41:45it is. So what is your radius? Your
41:47radius is 6. This is the radius for
41:50your question. And what were the
41:52coordinates of the center? By the way a
41:54, b, so here is your a and here is your
41:57b, here is a, b, so what was your
42:00target? Your target was number one,
42:03center, and number two, radius. So,
42:06what extra work did we do in this
42:07question? We applied the completing the
42:10square form. In the previous six
42:12examples, we didn't put in any extra
42:14effort because the whole square form
42:16was already given. The whole square
42:18form was already given. The bit of
42:20extra effort in this question is that
42:22we need a sense of completing the
42:23square. If we don't know the completing
42:26the square form, we can't solve this
42:27question. Now, let's look at a slightly
42:30better question than this. Part H:
42:32interestingly, questions of these exact
42:35same two types have also appeared in
42:37past papers. Now, G was easier than
42:40this. H is a little bit tough. Now
42:42let's tackle H. H is a little bit tough
42:44. Not too much, just a little bit tough
42:47. Clear, kid? Now I will guide you on
42:49how this will be applied. Okay, kid?
42:51Come on, let's get to it. Look at the
42:53next question. What is the next
42:54question? 2 x² + 2 y²-14x-10y-163 = 0
43:04, okay? We will look at this one quite
43:08carefully, please. Okay? Because this
43:10is more important. This question is
43:12more important. Number one, let's see
43:15again that the coefficient with the
43:17whole square must always be one. The
43:20coefficient isn't there. I mean, even
43:22if I take the x terms as common. Even
43:25if I combine the x terms together. Here
43:27, I have combined the x's. Put them on
43:29one side. Combined the y's together.
43:31There we go. Just like we did in the
43:33previous question. First x and y, the
43:34same thing happened here too. But I
43:36know that with x², with x, with y, it
43:39must be one. So step number one. What
43:41is the first thing I did? First, I took
43:44the two as common. So that at least the
43:47two is gone from the question. Clear,
43:49kid? I took the two as common. Here it
43:51is. Now just note my steps. x² + y²-
43:577x-5y-81.5 Equals to what did it become
44:04? = 0 it became. = 0, = 0, done. Now
44:11ignore this two. Now your question has
44:14converted into a completely simple,
44:16previous question. It has completely
44:17converted into the previous question.
44:19Come on, first I will copy it the same
44:20way. x² + y²-7x-5y-81.5 = 0, now look
44:25, now this thing has become the same.
44:31Now this whole thing and this whole
44:32thing have become the same. Now the
44:34work that was done there, separate the
44:36x terms first. Separate the y terms. I
44:39am doing the exact same thing here. So
44:41what did I do? First, separate the x
44:43terms. There you go. Then, similarly,
44:46separate the y terms. There you go. And
44:49then what is left here? -81.5 is equal
44:52to what? 0 = 0 Now what will you apply?
44:58Now you will apply completing the
45:00square. Because we know that the
45:03required form we need uses a whole
45:05square. So what will you have to use
45:07now? Now you will have to use the
45:09completing the square method. Now come
45:12on, child, let's go slow. Let's look at
45:14the completing the square method once
45:15again. Look child, in completing the
45:17square, see, this is x squared. Okay?
45:19We started. This is a squared. There is
45:22a minus sign in the center. This means
45:24there will be a minus sign in your
45:26formula too. Now what formula do we use
45:29? 2*a*b, so we know how to find b? We
45:33have already studied this. How do we
45:35get b? b is always the coefficient of x
45:38/2, coefficient of x/2, so from here I
45:41found out that b is 7/2. So I
45:44remembered that we used to add and
45:46subtract the whole square of 7/2. Is it
45:49clear, child? I did exactly the same
45:51work as the previous question. I
45:52applied the whole square formula, the
45:54completing the square form. Just like
45:56the previous question. Target achieved.
45:59Plus, now we will do the same work for
46:02y as well. For this too, we have to
46:04apply completing the square on this. So
46:06here, completing the square will be
46:07applied to this too. There you go. Now
46:09tell me, what is y? Here it is. This is
46:11y squared. In this too, what is the
46:13sign in the center? Minus. So there
46:14will be a minus in the formula. So what
46:16used to be in the center of the formula
46:18? 2*a*b, so what is its coefficient? 5,
46:21so what did we use to do? We used to
46:24divide the coefficient by 2. This is
46:25its b. So then we know we used to add b
46:28squared and we used to subtract b
46:30squared too. Finally, in the exact same
46:32way, we applied completing the square.
46:34Just like we did for x, the same work
46:37for y, the work for x was the same work
46:39for y. There, I did the same work for
46:43both and copied -81.5 as it is. There
46:47you go. Is it clear, child? There is no
46:49difference between these two methods.
46:50Obviously, I applied completing the
46:52square to both. So tell me, what will
46:54this whole thing become when completed?
46:56Okay? Now, if you make this a little
46:58bit easier. Let's make it a little
47:00easier. Here you go. So, if I talk
47:02about this bracket, what does this
47:03whole thing become? This entire thing
47:06becomes (a-b) ². Here you go. (a-b) ²
47:10because what is this? a²-2ab + b², so
47:14we know what its closed form will be.
47:18Okay? What will be its closed form? We
47:21have already studied this. So, what
47:23will be its closed form?(a-b) ² is
47:25right here. (a-b) ² is right here. And
47:30what will be the square of 7/2? 49/4,
47:34here you go, this will be 49/4, here
47:37you go. Is it clear, son? We have
47:41packed and closed this one thing. Next,
47:43next, now tell me, what is this entire
47:46thing? This is the same thing too. a²-
47:482ab + b², now pack this one up in the
47:52same way. So, when this closes or
47:55completes, what will it become?(y-5/2)
47:58² and copy 25/4 as it is, here you go,
48:0325/4 copied as it is, here it is, and
48:08-81.5 = 0, copy as it is, here you go.
48:14Finally, let's make it a bit easier now
48:16. (x-7/2) ² copied the same. Here you
48:20go. This is copied the same. Next, what
48:23is this? It's a plus sign. (y-5/2) ²
48:27copied the same. (y-5/2) ² copied the
48:32same, here you go. (y-5/2) ² copied
48:36the same. Now, shift everything else
48:38remaining to the right side. This
48:41becomes 49/4 + 25/4 + 81.5. This is now
48:47shifted for you. Done, so now look here
48:51, 49/4 + 25/4 + 81.5, this becomes. 100
48:56, this becomes 100, so after
48:58simplifying it finally, it becomes x-7.
49:01/2 squared, this becomes (y-5)/2
49:04squared, (x-7)/2 squared, (y-5)/2
49:07squared = 100 = 100, now your question
49:11has been converted into the same form
49:14we used in the previous six questions,
49:18the form which is called standard form,
49:21the form which is called standard form.
49:25Match it with the standard form. When
49:28you match it with the standard form,
49:30you know here is x, here is x, here is
49:31minus, here is minus. So from here it
49:34matches that a becomes what? a becomes
49:377/2, similarly y-b, so b becomes what?
49:42b becomes 5/2, r squared becomes what?
49:46r squared becomes 100, so then what is
49:48r? r finally becomes 10. So that means
49:51what are the coordinates of the center?
49:54So what coordinates of the center did
49:55we find? The coordinates of the center
49:58are 7/2, 5/2. Here you go, 7/2, 5/2.
50:01There you go. What is the square of the
50:04radius? The square of the radius is 100
50:06. So then, what will just the radius be
50:08? The radius will just be 10. This is
50:10your radius. Clear, son? This is your
50:13radius. This was your final target. So
50:16today, in these last two questions,
50:17what new work did we do? In this, we
50:19applied completing the square. I also
50:22explained in the introduction today
50:24that quadratics must be known before
50:26this topic. That is why we studied
50:28quadratics first. Clear, son? We
50:30studied quadratics first. Done. Now,
50:33the introduction to this center form is
50:35complete today. Now, let me explain a
50:38little further what we need to do after
50:41this. Okay, son? Today we studied this
50:44center form. Now, let's try one or two
50:45more questions. Then I will involve
50:48circle properties in this. Look at this
50:50, son, look at this question. Now, what
50:52hint is given in this question? If you
50:55note, then in this question too, what
50:58things do you already have? Note it.
51:01What things do you already have in this
51:03question? Number one. The center is
51:06already given in this question, and the
51:09radius is also already given. The
51:12circle's center is already given, and
51:14the radius is also already given. Done,
51:16son, there is the center and the radius
51:17too. What is your target? Equation of
51:20each circle. Let me revise the equation
51:22once more. Then we move ahead. So, in
51:25the whole day today, we studied only
51:26one thing. What did we study? What the
51:28standard equation of a circle is. What
51:31thing did we study? We studied what the
51:33standard equation of a circle is.
51:35Standard equation of a circle. Okay? So
51:40, we studied this standard equation of
51:41a circle. I will revise once more what
51:44the standard equation of a circle is.
51:48Now see what we will do. Now what is
51:51its equation? x-a² + y-b² x-a² +
52:01y-b² = radius squared. How do we read
52:05this equation in the language of math?
52:08We read this equation in the language
52:10of math. Standard equation of a circle.
52:12Obviously, we have only studied this
52:14thing so far. Okay, son? So far, we
52:16have only studied this. Now I am moving
52:19ahead. Okay, son, so what did we study
52:22in the previous question, what the
52:24standard equation of a circle is? x-a²
52:26+ y-b² = r² and we already know what
52:30the coordinates of the center are in
52:34this. The center has coordinates a, b,
52:37and what is the radius r? r represents
52:40the radius. And we have already studied
52:42what x and y are, right? x and y are
52:45any point on the circumference. Now,
52:49look at all these parts; this is also
52:51an average question. In all these
52:54questions, center and radius are
52:57already given. What is your job? Your
53:01job is just to plug in the values and
53:03find the equation. Here it is. (x-a) ²
53:06and (y-b) ² right here. Now tell me,
53:10what is the center? The center has
53:12coordinates 0, here it is. So here it
53:14becomes (x-0) ² and what is this?(y-0)
53:18². Okay? What happens here?(x-0) ² +
53:23(y-0) ² = 8², meaning we simply put 0
53:28in place of a and b. And we put 8 in
53:32place of r. So, if I simplify this
53:35further, it becomes x² + y² = 64,
53:39that was your target. What was required
53:42, kid? We required the equation of the
53:44circle. So it was a very simple
53:45question. A very simple question. Clear
53:48, kid? So what did we simply do in this
53:50? We simply plugged in a, b, and r. We
53:53plugged in a, b, and r. Look, this
53:55equation is what we call the equation
53:56of a circle. (x-a) ² + (y-b) ² =
54:00radius squared. Now tell me, this is
54:03the center. So that means this is a and
54:05this is b. And this is the radius, so
54:07this is r. Now you just plug in the
54:09values. So now tell me, what is its a?
54:11Its a is 5, and what is the b in this
54:13question? -2, so minus-minus will
54:16become +2, and what is this? Radius.
54:18The radius is 4, so if I simplify this
54:21further, what will this equation become
54:25?(x-5) ² Next, (y + 2) ² = 16; this
54:32type is your standard equation of a
54:35circle. This type is the standard
54:37equation of a circle. So, there was
54:39nothing extraordinary about this
54:41question. We just plugged in A, B, and
54:44R. We know what the coordinates of the
54:47center are. We know that the
54:49coordinates of the center are A and B.
54:52The coordinates of the center are A and
54:54B. Look at part C of this. Look at part
54:57C of this same question. In part C, we
55:00know what the equation of a circle is.
55:02(x-a) ² (y-b) ² = radius squared.
55:08This is the equation of the circle. Now
55:10tell me, what is the center in the
55:12third case? a, b, and what is the
55:14radius? The center is a equals -1, b is
55:18three. Then just put in the values. In
55:20place of a, put -1, and what will you
55:23put for b? You will put three in place
55:26of b. -1 in place of a, and three in
55:28place of b. What will the radius
55:29squared be? It will be 7². So if I
55:33simplify this, what answer will be
55:35derived? x + 1 squared, y-3 squared = 7
55:39, this is the final answer derived for
55:42your question. This is the final answer
55:47derived for this question. Now we will
55:50try part D of this. We will try part D
55:53of this. Look son, now we know that we
55:56did the same work in all these
55:58questions. x-a² + y-b² = radius
56:02squared. Now tell me, what is the
56:06center? The center is A and B. Who is
56:08this? This is the radius. Now tell me,
56:11what will we finally do in this
56:12question? Finally, we will plug in A
56:15and B. What is A? A is your half. So,
56:18we replaced A with half. Because we
56:20know what A and B are, right?
56:21Coordinates of the center. And what is
56:23B? Note it down carefully. b is -3/2.
56:26So when you put -3/2 here, then what
56:29will minus minus become? Minus minus
56:31will become plus. So this will become y
56:34+ 3/2 squared equals the radius squared
56:38. What radius do you have? 5/2. What is
56:42the radius? 5/2, meaning there was
56:44nothing new in this question. Just that
56:46the coordinates of the center were
56:48given in fraction form. If I simplify
56:51this further, it becomes x-1/2 squared,
56:54x-1/2 whole squared, if I simplify this
56:57further, y + 3/2 whole squared, there
57:00you go, equals what? It equals 25/4,
57:03your target is achieved. Meaning, what
57:07was there here? Here you had no extra
57:10effort. Meaning you were already given
57:12the center's coordinates. And you were
57:14also already given the radius. So
57:15technically, what did we do in this
57:17whole basic question? In this basic
57:20question, we only and only plugged in
57:22values. Clear, son? We only and only
57:25plugged in values. Okay? So today we
57:27saw the introduction of the standard
57:28form of a circle. In the next class,
57:30InshaAllah, we will start practicing
57:32more of this. Okay, son? Allah Hafiz.

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