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AS Mathematics 9709 | Coordinate Geometry 10 | Relationship between Point & Circle | Raja Shafqat — Transcript

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0:08Assalamu Alaikum, beta. Finally, today
0:11we are starting the second point of the
0:13circle. Well, this point appears in the
0:15paper for one or two marks. One or two
0:18marks. What is the name of the point?
0:20Relationship between point and circle.
0:22What are the possible relationships
0:25between any point and a circle? That is
0:28, we have to check whether the point
0:31lies inside, on, or outside the circle.
0:34I will explain what these three
0:36possibilities mean. Look, I will guide
0:38you one by one. It's a small point, by
0:40the way. It only comes for one mark in
0:42the paper. Okay? It comes for one mark.
0:45That means it must be a very basic
0:46point. Suppose you have a circle. Okay?
0:50Suppose I have this circle. This is a
0:52circle. I just plotted a circle
0:54randomly. Okay? I plotted the circle
0:57just randomly. Come on, now I will copy
0:59this two or three times so that I can
1:01explain all the conditions one by one.
1:04Okay, beta? I copied the same circle
1:06three times. Number one, the first
1:09condition: Math says that if any point
1:12is on the circle, what happens? If I
1:15tell you that I have a circle. Let's
1:18align all these circles, I will tell
1:21you in a bit more detail. Let's do one
1:24thing. Let's fix this. The name of the
1:28center of all these circles is C. Okay,
1:30what did we do? We named the
1:32coordinates of the center of all these
1:35circles C. C is the center of the
1:38circle. Done. Come on. And the point,
1:41we have named that point P, the point
1:44we are checking to see what its
1:47relationship is with the circle. Let's
1:50call the coordinates of that point P (x
1:53, y). x y, now what is our task? To
1:57check the relationship. What are the
1:59possible relationships between a circle
2:01and a point? Number one, situation
2:03number one. The first condition. Look
2:06at the first condition; math says that
2:09if you have any point on the circle. If
2:12you have to show the examiner that
2:14point P is on the circle. This is your
2:17point P. Suppose this is your center of
2:20the circle. This is your center of the
2:22circle. Come on, it must be somewhere.
2:23There must be some point as the center
2:25of the circle. So if you have to prove
2:27that point P is on the circle. Point P
2:29is on the circle. So, it's a very
2:32simple logic that the distance from the
2:36center to point P must be equal to what
2:40? That distance must be equal to the
2:42radius. Clearly, equal to what should
2:46it be? It should be the radius. Is that
2:48clear, son? That distance must be equal
2:50to what? It must be the radius. What is
2:53C? C is the coordinates of the center.
2:54What is P? P is a point on the
2:57circumference. A point on the
2:59circumference. And what is C? C is the
3:02coordinates of the center. C is a point
3:04. Meaning, C is also a point and P is
3:06also a point. C is the center of the
3:07circle. And what is P? P is a point on
3:10the circumference. So, let's name the
3:12radius of this circle as r. Suppose the
3:15radius of the circle you have, let's
3:17consider it as r. Let's assume the
3:20radius to be r. So if a point is on the
3:22circle, if any point is on the circle,
3:25it means the distance from the center
3:27to the point must be equal to the
3:30radius, right? It's a very simple
3:32condition. That is why this point often
3:33appears as a one-mark question in exams
3:35. Situation number two. Situation
3:38number two. Suppose you have the same
3:41circle and the center of this circle is
3:43C. You need to prove that point P is
3:45outside the circle. Let's say you have
3:47to prove that there is a point, and
3:49that point is outside the circle. Let
3:51this be our target. We have to prove
3:53that our point is outside the circle.
3:56And what have we considered for the
3:57coordinates of that point? We have
3:59considered the coordinates of the point
4:00as (x, y). Okay?(x, y), I need to show
4:03the examiner that point P is outside
4:05the circle. So this is also a very
4:07simple matter. If we use even a little
4:09bit of common sense. If we use a little
4:12common sense, we have to prove that
4:14point P is outside the circle. Then
4:16math says that if a point is outside
4:19the circle, then the distance of that
4:21point from the center will be greater
4:23than the radius. Obviously, the radius
4:26is just this small gap, right? The
4:28radius is from the center to the
4:30circumference. It is from the center to
4:32the circumference. So if a point is
4:34outside the circle, then the gap
4:36between P and C will obviously be more
4:38than, greater than, the radius. So that
4:41means our target will be to find out CP
4:45. Who is CP? The distance between the
4:48center of the circle and any point that
4:50you need to check. If the distance of
4:52that point is greater than the radius.
4:55If the distance of that point is
4:57greater than the radius, then it is
4:58confirmed that point P is outside the
5:00circle. So what will we call this
5:02condition? In this condition, we will
5:05say that point P is outside the circle.
5:08Point P is outside the circle. Okay?
5:11It's a small one-mark condition. Point
5:13P is outside the circle. What will we
5:15say in the first condition? In the
5:17first condition, we will say that point
5:20P is on the circle. Point P is on the
5:24circle. Point P is on the circle. In
5:27the second condition, we will say that
5:28point P is outside the circle. Third
5:31and last condition, third and last.
5:33Suppose you have a circle and you
5:35already know its center; its center is
5:37C, here is its center C, and you have
5:39to prove that there is a point, point P
5:42, which is inside the circle. Come on,
5:45this time show that your point P is
5:48inside the circle. Okay? What have we
5:51considered the coordinates of point P
5:52to be? X Y. We have considered the
5:55coordinates of point P as X Y. And what
5:57is the center of the circle? The center
6:00of the circle is A B, what is the
6:01center of the circle? The center of the
6:04circle is A B, we have to prove it. We
6:06have to prove that point P is inside
6:09the circle. So it is a very simple
6:11matter. If point P is inside the circle
6:13, then the distance between point P and
6:16C must be less than the radius. That
6:19distance must be less than the radius.
6:21Because the radius, what is the radius?
6:23Radius is the distance from the center
6:25to the circumference. From the center
6:27to the circumference. Clear? So it
6:29means that your distance between P and
6:32C, between the center and the point,
6:34between the center and the point. If
6:37the distance between your center and
6:39the point is less than the radius. If
6:41the distance between the center and
6:43point P, which you are checking, is
6:45lesser than the radius. Then what do we
6:48call this condition in the language of
6:49math? In the language of math, we say
6:52that point P is point P is inside the
6:54circle. But is inside the circle. So,
7:00this is a small point that comes in the
7:02exam for one or two marks, that comes
7:04in the exam for one or two marks. So,
7:06there are three possible relationships.
7:08You see, what do you have to do in all
7:09three cases? In all of them, you need
7:12the distance between the center and
7:13point P. In all of them, you need the
7:15distance between the center and point
7:17P. Now, if it is equal to the radius,
7:19then the point is on the circumference.
7:22If it is greater than the radius, then
7:23the point is outside the circumference.
7:24Outside the circle, if it's less than
7:27the radius, it will be inside the
7:29circle. Clear, dear? So this is a small
7:32point that comes in the exam for just
7:33one mark. Clear, dear? What do we have
7:36to do in every condition? What do we
7:38have to check in every condition? What
7:41is there to check? The relationship,
7:42whether the point is outside or inside.
7:44This is just a one-mark point. Let's
7:46check what particular type of question
7:49will be in the exam. What will we have
7:51to do? Okay, dear? Let's check. Look at
7:53number one. We have to check if the
7:56point is on, inside, or outside. It's
7:58the same statement. The point is
7:59already given in all questions. The
8:01center is already given in all
8:02questions. The radius is already given
8:04in all questions. Everything is given
8:06in all questions. Clear, dear?
8:09Everything is given. So how will our
8:10one-mark question be solved? Come, let
8:13me start with part one of A. Here you
8:15go. So I will start with part one of A.
8:17Now you tell me what your point P is? 1
8:207, here is point P. What is your center
8:23? By the way, the center is -2 3, which
8:26is already given. So what will we do?
8:28First of all, we will apply the
8:30distance formula. We checked what the
8:32distance between P and C is. This is a
8:34one-mark point, by the way, because
8:36everything is already given. (x2-x1) ²
8:38+ (y2-y1) ², so this distance came out
8:41to be five. Look at what your radius
8:44was given as. Your radius was also
8:46given as five. Your radius was also
8:48five. So the gap between P and C became
8:50exactly equal to what? It became
8:52exactly equal to the radius. So look at
8:54what the situation is. The situation is
8:56basically this. If we check this
8:58condition, then to be on the safe side,
9:00I will show it to you. Look at what the
9:02condition is. Here you go, this is the
9:05situation, by the way. Okay? This is
9:07the situation, by the way. If I
9:09summarize for you what happened in this
9:11question? Then we will say this
9:13happened in this question. Clear, dear?
9:15So what do we simply have to do? We
9:17simply have to apply one condition in
9:19this question, which is to check the
9:21distance. Whose distance do we have to
9:23check? We basically have to check the
9:25distance. We basically have to check
9:27the distance between the point and the
9:29center. Now some condition will match,
9:31like this condition matched. Finished.
9:33That is why this point is for one mark.
9:35Only one mark. Okay? Look at part two
9:38of a here. Look at part two of this
9:40question's a. That's why this topic
9:42comes for one mark in the paper. Now
9:43you are already pointed. I pointed 2 -1
9:46, here it is. And what is the center?
9:48The center is -3, here it is. Now see
9:51what the distance between P and C is.
9:53Apply the distance formula. x2-x 1²,
9:56meaning we have already done the
9:58distance formula in coordinate geometry
10:00. -3 -2 -5² 3 -1 4² 25 + 16 41 41,
10:06there you go. So here too, the distance
10:10of p and c came to 41. The interesting
10:12thing is, what radius were you given in
10:14this question? The radius you were
10:16given was 41, here it is. So your
10:19answer was derived the same. If the
10:21answer was derived the same, it means
10:23the work done in the previous question
10:25was exactly the same work done in this
10:26question. This question also has the
10:28same situation. Look, this question
10:31also has the same situation. I will
10:34repeat it again. I copied those
10:36sketches so that time wouldn't be
10:38wasted. Okay, son? So time isn't wasted
10:40. I just had to give you an idea. This
10:42is a one-mark question. It is a
10:43one-mark question. Done. So we gave an
10:45idea to the examiner. So how is this
10:48one-mark question proven? You just
10:50always have to find the distance.
10:51Whether it is equal, less, or more. So
10:54some condition will definitely match.
10:56Look at part one of b of this same
10:59question. Look at part one of b of this
11:01same question. B part one. B part one.
11:06Okay, son? Come on, look. Part one of
11:08B. We are doing only one thing.
11:10Continuously every question. Okay, look
11:12at number one. Now what will you do?
11:14Now look, you already know point P.
11:16Look, what is point P in B part one?
11:18Point P is -1 1, what is the center of
11:21the circle? 3 6, this is the center of
11:23the circle. 3 6, okay, what is the
11:26given radius? The radius is already
11:27given as five. All of this is already
11:29given. Now what is our job? That we
11:32have to find the distance between P and
11:34C. We have to find the distance between
11:35P and C. Okay? As always in every
11:39question, when we found the distance: (
11:413) 1 4², (6) 1 5². So this will come
11:45to 41, there you go, this came to 41,
11:47it's a one-mark question, by the way.
11:5041, this is how your answer was derived
11:51. Is it clear, beta? 41, your answer is
11:55derived. The answer 41 is derived, okay
11:58, now look at what 41 is? 41 will
12:02obviously be 6 point something, okay.
12:06The root of 36 is 6. So this is more
12:08than 36. So what will this be? It will
12:11surely be greater than 5. Is it clear,
12:12beta? So from here I found that the
12:15distance between B and C that I got is
12:16greater than the radius. So what does
12:19it mean to be greater than the radius?
12:21Which condition is being satisfied? It
12:24means that this condition is being
12:26satisfied. Look, this condition is
12:27being satisfied. I will copy this
12:29condition. Look here beta, this
12:31condition of yours is matching. I am
12:33showing you the diagram so that if a
12:36student revises, okay, they can match
12:38from here what the teacher had
12:40explained. Clear beta, what the teacher
12:43had explained. That is why I am showing
12:44the sketch. Otherwise, this topic is
12:47for one mark, so you can connect it
12:48with the diagram. Connect it with the
12:51diagram. So the gap between your point
12:53P and C was more than the radius. The
12:56radius was five. Your distance is
12:57greater than the radius. So then what
12:59does this mean? It confirms that your
13:03point is outside. Look at the last and
13:06final point, part two of this. This has
13:08appeared twice in past papers, but only
13:10for one mark each. Here you go. Now,
13:13what is given in this question? Point P
13:15is given as 2, 1 and what is the center
13:18given? The center is given as 5 and -1,
13:21here it is. And what is the radius
13:23given? The radius is given as 7. Let's
13:24check what the condition is. What is
13:26the situation? As always, we looked at
13:28the distance between P and C, and for
13:30distance, the distance formula will
13:31always be applied. 5-2 is 3 squared,
13:34-1-1 is -2, -2 squared, 9 + 4 is what?
13:3813 9 + 4 is what? 13, no need to even
13:43use a calculator. The root of 9 is 3.
13:46So the root of 13 will be 3 point
13:48something. It will be 3 point something
13:49. So the gap that you got is less than
13:52the radius. By the way. This gap of
13:54yours is less than the radius. By the
13:56way. Is it clear, beta? If this gap is
13:58less than the radius, the point will be
14:00inside, right? Meaning this situation,
14:02this condition of yours is matching.
14:05This condition is matching. Look at
14:07this. Now you see it again. So what did
14:09I do? I have copied all the same
14:11sketches here that I explained to you
14:12in the summary earlier. Clear, kid? So
14:15that you can graphically see the impact
14:17of what I explained. What was to be
14:19done? These were small questions,
14:21nothing very extraordinary, kid. Clear?
14:23This is a one-mark concept. By the way.
14:26If we summarize the concept, it only
14:28comes for one mark in the exam. Okay,
14:30kid? It only comes for one mark. Here
14:34you go. So, I have shown you all the
14:36same diagrams. Clear, kid? So finally,
14:39in this question, this was just a small
14:41one-mark point. This point of yours is
14:43completed. Okay, kid? This point of
14:46yours is completed. Nothing
14:47extraordinary. Done. This was your
14:50target. Target achieved. Okay, kid?
14:53This was your target. Target achieved.
14:55This was a one-mark point. I will
14:57explain it once again that if a point
14:59is outside the circle, its distance
15:01will be greater than the radius. The
15:03gap between the center and that point
15:04will be greater than the radius. If a
15:07point is inside, the distance between
15:08the center and the point will be less
15:10than the radius. If a point is exactly
15:13on the circumference, the gap between
15:15the point and the center will be
15:16exactly equal to the radius. It will be
15:19exactly equal to the radius. Clear, kid
15:21? So, what was the name of this small
15:23point? The name of this small point was
15:26the relationship between a point and a
15:29circle. There are three types of
15:31relationships. If I summarize, either
15:34the point will lie on the circle,
15:36inside the circle, or outside the
15:38circle. What will happen with one of
15:40the three conditions? One of the
15:42conditions will always be satisfied.
15:44That's it. This was a one-mark point.
15:46Now, what I am doing is a bit technical
15:48. Okay, kid? This is a slightly
15:51technical condition. That's why it
15:52requires a little effort. Condition
15:54number three. Situation number three.
15:56Now, first you read it for two minutes.
15:59Now we have to check what possible
16:01relationships can exist between two
16:03different circles. Okay? What possible
16:05relationships can there be? Now I will
16:08explain to you what situations can
16:10occur. Okay? Read again, if the
16:13examiner gives me two circles. I don't
16:16know if they are of the same size or
16:17different sizes, I will explain that
16:19too. So, what is our target? To check
16:21the relationship between two circles.
16:22What are the possible relationships
16:24between two circles? Done, kid? Now I
16:27will explain it to you one by one. One
16:29by one. Now I will show you different
16:31situations. Then obviously, after a
16:32while, I will have you practice this.
16:34Situation number one or case number one
16:36. First situation. Okay? First
16:39situation. Well, the first situation is
16:41called disjoint circles in the language
16:43of math. Case number one is called
16:46disjoint circles in the language of
16:48math. Okay? Well, let me first explain
16:51to you what we are trying to do. Okay?
16:53I will show you different situations
16:55first. So, the name of the first
16:57condition is disjoint circles. Disjoint
17:00means two such circles that have no
17:02relationship. So let's start with
17:04situation number one. The name of case
17:07one is disjoint circles. Okay? Let me
17:11start with case number one. Disjoint
17:13circles. Okay? Then, God willing, I
17:15will explain all the following cases to
17:17you one by one. Is that okay, dear? I
17:19will explain all the following cases to
17:21you one by one. So, the first case we
17:23are starting with. Okay? Let me copy it
17:26like this. The first case we are
17:28starting with, the name of this case is
17:30disjoint circles. First, I will show
17:33you the graphical impact once so that
17:35you start to understand. Then after
17:37that, we will decide what to do in the
17:40exam. After that, we will decide what
17:42to do in the exam. The name of the
17:44first situation is disjoint circles.
17:46Come on. Math says if two circles are
17:48disjoint. Whether the size is big or
17:51small, it does not matter. Is that okay
17:52, dear? Now first see the situation,
17:54what the examiner will tell us. Okay?
17:57Now suppose I have a circle. Okay?
17:59Suppose one circle is like this and one
18:01circle is like this. I am showing you
18:03two such examples. Is that okay, dear?
18:05I am showing such examples. This larger
18:07circle, suppose this is its center.
18:10Okay? This is its center. And this is
18:12suppose the smaller circle's center. I
18:14am just showing a random example. Okay?
18:16Larger, smaller. What did I do just
18:19randomly? I plotted two circles. I just
18:21randomly plotted two circles. Okay? Now
18:24I will show you. These are disjoint.
18:26Meaning they are not intersecting with
18:27each other. They are not intersecting
18:28with each other. Is that okay, dear?
18:30Let's name the center of this circle C1
18:32. Come on, for the center of this
18:35circle, I will tell you to explain that
18:37its center is C1 and let's name its
18:39radius R1. Come on, let's do labeling
18:42to differentiate. Let's name them. Its
18:46radius is R1. Let's assign a name to
18:49the center of this circle. Its center
18:51is C2. And let's name its radius R2.
18:57The radius's name is R2. Now I need to
18:59prove to the examiner that these two
19:01circles are disjoint. So how will I
19:03justify that? All right, son? Now, look
19:05closely. Its center is C1 and its
19:08center is C2. So how will we prove it
19:10to the examiner? We will take the
19:11distance between these two points. Tell
19:13me one thing. This is your R1. This is
19:16your R2. If two circles are disjoint,
19:18the logic is very simple. The gap
19:21between the centers of these two
19:22circles. If I have to prove that these
19:25two circles are disjoint, then the gap
19:27between their centers. Right? Look
19:30again, what do we need help from? We
19:33need help again from the distance
19:34formula. From what should we take help?
19:36We need help from the distance formula.
19:38Look very closely. Now, how will we
19:40prove that two circles are disjoint?
19:42Math says that you look at the gap
19:44between their centers. If the gap
19:47between their centers is greater than
19:49the sum of the radii. Then the circles
19:53are disjoint. Then the circles are
19:55disjoint circles. Look at the condition
19:58again, son. What condition did I tell
20:00you? That the gap between the centers
20:02of those two circles, look closely, if
20:04I apply the distance formula to them,
20:06we will be given the centers and the
20:08radii. We applied the distance between
20:10these two centers. Found the length
20:12between these two centers. Found the
20:14distance. If that distance is visible.
20:17R1 is up to here. R2 is up to here. So
20:20it contains something extra, right? It
20:22has some extra length in its path. So
20:25that means the circles are disjoint
20:27circles. So I will show you different
20:30situations. Then finally we will apply
20:32it. Then we will see its example. Clear
20:34, son? This is the first condition. It
20:35is called disjoint circles. Disjoint
20:37circles. Done? This is the first
20:39condition. Well, disjoint means
20:41non-intersecting circles. These are
20:43those circles that do not intersect.
20:45All right, son? Disjoint means they are
20:46not intersecting. Second condition. The
20:49second condition is called intersecting
20:51circles. Intersecting circles. I am
20:54showing another situation that if the
20:56circles are intersecting, then what
20:57will happen? All right, son? Then what
20:59will be the condition? Second situation
21:02number two. The first task we did was
21:05disjoint. Now, the second task we are
21:08doing, the second task we are doing is
21:10about intersecting circles. Okay? The
21:12second task is about intersecting
21:13circles. Now, look, what will we do in
21:15intersecting circles? Situation number
21:17two. Now again, I will plot two circles
21:20and consider their centers C1, R1, and
21:22centers C2, R2. Now I have to show the
21:25examiner, suppose this is one circle.
21:27Okay? This is the circle. Its center is
21:30C1 and radius is R1. And this is
21:32another circle. Let's say it's like
21:33this. I have plotted one more small
21:35circle. Okay? Look, I will mark its
21:38center to show you. Suppose this is
21:41your larger circle and the radius of
21:43the larger one is R1. Suppose this is
21:47your smaller circle and its center is
21:50C2. Let's name the radius R2, C2, R2;
21:53so this suppose is C1. This suppose is
21:55C2. Its radius is R1. We labeled its
21:59radius R2 so that we can differentiate.
22:02Now, what will we do in this condition?
22:04You will join the centers of both. You
22:07will join the centers of both and look
22:08at the distance between these two
22:10centers. You will look at the gap, that
22:12is, you will look at that same gap
22:14again. The distance of C1 and C2,
22:16distance between C1 and C2. Distance
22:20between C1 and C2. Is it clear, son? We
22:23will look at the C1-C2 gap. Now tell me
22:25one thing. Where does R1 reach? Up to
22:27here. This is R1. Where is R2? From the
22:31center up to here. So that means this
22:34whole thing is R1. Up to here is R1. Up
22:37to here is your R2. Okay? So, when I
22:40add these two radii, their sum will be
22:43greater than the distance between the
22:45centers. Their sum will be greater than
22:48the sum of the radii. Why will it be
22:50greater? Greater than what? Because of
22:53overlapping. Okay? Look, listen, now
22:56one thing, let me explain it again. I
22:58made a small mistake. What will this be
23:01? It will be less. Why will it be less?
23:04I will tell you again. Why will the sum
23:06of the radii be greater? The sum of the
23:08radii will be greater. The value of the
23:11sum of the radii will be greater. Why
23:12will it be greater? Because of
23:15overlapping. Because of overlapping.
23:19Because of overlapping. What does this
23:21mean? What is the meaning of
23:22overlapping? Let's listen again.
23:24Brother, when you talk about the
23:25distance between two centers, that is
23:26simply this gap, right? Right. This is
23:28the distance between two centers. You
23:30will be given the coordinates of the
23:31centers. If you apply the distance
23:33formula here, you will get the distance
23:35. But when you add the radii, look,
23:36what is the gap up to here? R1 R1, and
23:40what is the gap up to here? R2, so when
23:44you added R1 and R2, this was the extra
23:46distance. Because of this extra
23:49distance, because of the overlapping,
23:51what happened to the sum of the radii?
23:54It became greater. What was it earlier?
23:57The distance between the two centers
23:58was greater. It was greater than the
24:01sum of the radii. Now, the distance
24:02between the two centers is less than
24:04the sum of the radii. Why is the sum of
24:06the radii greater? Because of the
24:08overlapping. Overlapping means this gap
24:11is R1. This gap is R2. So now look,
24:14this middle portion has been counted
24:16twice. In R1 as well, and in R2 as well
24:19. In R1 as well, and in R2 as well.
24:21That's it. So, because of the
24:24overlapping, the distance between the
24:26two centers of intersecting circles
24:28will be less than the sum of the radii.
24:31The sum of the radii. Is that clear,
24:33son? Now let's look at a third
24:35condition. Okay? Situation number three
24:38. Let me show you one more condition.
24:40Then we will finally look at the sums
24:41for all of these. Third condition.
24:43Let's read the third condition in the
24:45language of mathematics. Tangent
24:46circles. Tangent circles. That we have
24:51to prove that two circles are tangent
24:53to each other. Two circles are tangent
24:56to each other. Situation number three.
24:58I am showing this graphically first so
25:01that your mind can get set on what it
25:04technically or graphically means. What
25:08does this mean technically or
25:10graphically? Once we understand this.
25:12The situation we are looking at now,
25:14this third condition is called tangent
25:16circles. Tangent circles. Now let's
25:19check, tangent means to touch. Tangent
25:22means to touch. Clear, son? Now look,
25:24let me show you a small example so that
25:26you can understand this. Suppose you
25:28have this one circle and another circle
25:30. Here are two circles. Okay? These are
25:34two circles. Suppose you have this kind
25:36of condition. Just giving you an idea,
25:37by the way. Now, for this larger circle
25:41, you can name its center. Its center
25:44is, suppose, C1 and the radius is R1.
25:48Center C1, radius R1, and this smaller
25:52circle, smaller, okay? This smaller one
25:54, you can name the center of this
25:56smaller one C2. The center of the
25:59smaller one is named C2 and let's name
26:02its radius R2, so this situation is
26:04quite clear. All right? If you
26:06understood the previous things, then
26:07this situation is quite clear. Tell me
26:09one thing. Now, if I take the distance
26:11between these two centers, what will
26:13the condition be? If I take the
26:15distance between these two centers,
26:16what will the condition be? Oh, by the
26:19way, what should be the name of this
26:21point here? The name of this point
26:24should be point of tangency. All right?
26:27If I were to ask what we should call
26:29this point mathematically? Then we have
26:31been calling this point the point of
26:33tangency before as well. The name of
26:35this point was point of tangency. Point
26:38of tangency. Here, both circles are
26:41touching each other. Touching means
26:43intersection at only one point. So what
26:45does math say now? This gap here is
26:47your R1. This gap here is your R2. So
26:50how will you prove that two circles are
26:52tangent to each other? Meaning both are
26:54touching each other at one point. The
26:55method to check this is also similar
26:58and quite simple. What will you do? You
27:01will take the gap between the centers
27:03of these two circles. All right? What
27:05will you do? You will take the gap of
27:07the centers of these two circles. Okay,
27:09child? That remains. Now you checked
27:11its distance, what is the distance
27:13between these two? Math says that if
27:16your distance is equal to the sum of
27:19the radii. If it is equal to the sum of
27:23the radii, then what is the condition?
27:25The circles are tangent to each other.
27:27It is a very simple matter. Look, here
27:29is C1 plus here is C2. So this C1 to C2
27:32gap, this whole gap is here. So this is
27:34exactly R1 + R2. Meaning there is no
27:36overlapping anywhere. So, the gap
27:38distance between the two centers is the
27:41same as the length of the two radii. So
27:44these are three situations. If you look
27:46at these three conditions, there is a
27:48small difference in them. All right? We
27:51saw disjoint circles which have no
27:53relation. Then we saw intersecting
27:54circles and what their relationship is.
27:56Then we saw the third condition, which
27:58is called tangent circles. When the
28:00circles touch each other. Tangents
28:03means to touch. Which point is the
28:05point of tangency? The point of
28:08tangency is the point where two circles
28:10touch each other. Two circles touch
28:13each other. Clear, child? Intersect is
28:16a general word. It could mean two
28:18points or even one point. But touch
28:20means only one. One-point intersection.
28:23So, what is this condition? Tangent
28:25circles. Tangent circles. Okay, now I
28:27will show you two or three more
28:29conditions. Right? I am showing you
28:31different possibilities. Now I am
28:33talking about circles within circles.
28:35Right? Now we will look at two or three
28:37conditions. Please watch closely. Now,
28:40the fourth condition. Within this
28:41fourth condition, there are two more
28:43conditions. The fourth condition is
28:45called circles within circles. Right?
28:47So first, let me start. Then I will
28:50tell you what possibilities exist
28:52within this. Right? So now we are
28:54looking at circles within circles.
28:57Right? This is called circles within,
28:58there was a topic like this in O-Level
29:00math too. They used to call it a subset
29:02. Circle within a circle. Right? Circle
29:05within a circle. Here is a circle
29:07within a circle. Okay, so what is this
29:09condition? Let me explain it first. Now
29:12there are some possibilities within
29:13this as well. I will tell you all the
29:15possibilities one by one; in O-Level
29:18math, circles within circles were also
29:20called subsets. Right? They used to
29:22call it a subset in O-Level math. Look
29:25at number one. Now, inside this, there
29:27are three or four conditions; suppose
29:29you have a situation like this, okay?
29:31Now I will show you three or four
29:33conditions, just focus a little. Here
29:35you go, let's start. Here, suppose this
29:37is a circle, this is a larger circle, I
29:40have taken a larger circle, okay. Let
29:42me do one thing; I will write something
29:45here to explain it so it becomes easier
29:47for you. For the larger circle, the
29:50bigger one, let's take its center as C1
29:52. Come on, let's do something here
29:55beforehand so that we get a little help
29:58. Right? Let me do one thing. Let me
30:00set things up in advance. Then we will
30:03start the topic. Right? Let me do this
30:05work in advance first. Then we will
30:06start the topic. Come on, look. Right?
30:09We have two circles. One will be larger
30:12, one will be smaller. Look carefully.
30:15So, I will do this, the larger circle,
30:17the larger circle. Then I will start
30:20the topic. Let's name its center. Let's
30:25name its center C1 and its radius R1. I
30:29just did that to differentiate them as
30:32one and one. And then there will be a
30:34smaller circle. Common sense says that
30:36when there's a circle within a circle,
30:38one will be larger and one will be
30:39smaller. So this means that if we talk
30:42about the smaller circle, what will we
30:44do in its case? In the case of the
30:47smaller circle, let's name its center
30:51C2 and its radius R2. There, now let's
30:55start our question. We locked in 1 and
30:582 for a minute so we remember what is
31:00larger and what is smaller. Now we
31:02start reading our question. Now look,
31:05there are three or four situations
31:06within this as well. Okay? First, I
31:08will show you the first condition. Then
31:10we will move forward slowly. Come on,
31:12let me show you all the possibilities
31:13one by one. There are four or five
31:15conditions within this too. Here is the
31:16first condition within this, situation
31:18one. This is the larger circle. Suppose
31:21this is the center of the larger circle
31:23. Here it is. Suppose this is the
31:25center of the larger circle. And we
31:26know what we have considered its radius
31:28to be, right? We have considered its
31:31radius as R1. One, so we considered the
31:35radius as one. Center C1, radius R1.
31:38Okay. Now in the same way, let me show
31:40you another circle. Okay? Let me show
31:42you a small circle. Here it is. Let me
31:44show you one more circle. Here is a
31:45small circle. Here it is. Plotting
31:46another circle. This is the smaller
31:48circle. This smaller circle will
31:49obviously have a center. Every circle
31:52obviously has some center, right? So
31:54the large circle was named C1 and this
31:57small circle is named C2. Situation
31:59number one. Now look at what the
32:01condition is in this. What is the
32:03condition in this? Now if I have to
32:05prove it, look, there are two or three
32:07options. The first option is that the
32:10radius of the larger, the radius of the
32:12larger, the radius of the larger one
32:14will be...Okay? Let's do one thing.
32:17Let's call the larger radius capital R.
32:18Let's call the smaller radius small r.
32:20There, let's make the larger radius
32:22capital. There you go, and let's make
32:24the smaller radius small r. Okay,
32:26larger, smaller. So the larger radius
32:29will obviously be greater than the
32:31smaller radius. I can justify this in
32:33two or three ways. Then I will tell you
32:35the best option. Look. This means that
32:37the radius of the larger circle is
32:39greater than the radius of the smaller
32:41circle. This is a very basic point.
32:43Look at this. This is the radius R of
32:45the larger, this is the radius of the
32:47smaller. Whether you look at it this
32:49way or that way, the radius of the
32:51smaller one is very small, very small.
32:53Clear? So, I can also show it like this
32:56: look, the center of the larger circle
32:58, what did we name it? C1. I can
33:02justify it this way too, and explain
33:06this point a bit better, that the
33:09distance between C1 and C2, look, here
33:12is the distance C1 C2, here is the
33:15distance C1 C2, here is the distance C1
33:19C2 plus R2, plus R2 C1. The distance C2
33:22plus the radius of the smaller circle,
33:25plus the radius of the smaller circle,
33:27the larger radius is greater than both.
33:29The larger radius is still greater than
33:31these two. Look closely. The larger
33:34radius is still greater than both of
33:36these. Listen again. Let's look at it
33:38again. This gap here, what is this gap?
33:40C1 + C2, what is this? + R2, so you see
33:44what the full R is? The full R is from
33:47here to here. So R is still greater
33:49than this. Therefore, the distance
33:51between two centers plus the radius of
33:53the smaller circle is still less. The
33:55radius of the larger circle is still
33:57greater than both of these. The radius
33:59of the larger circle is still greater
34:01than both of these. So if we talk
34:03generally, we can also say, look, that
34:04the larger radius is greater than the
34:06smaller one. It is clearly visible. The
34:09larger radius is this whole thing. The
34:10smaller one is this or this. But if we
34:12speak more precisely, we can also say
34:14that the distance between two centers
34:16plus the radius of the smaller circle.
34:18This is still less than the total gap
34:20up to the circumference. So the radius
34:22of the larger circle is greater than
34:24the distance between two centers plus
34:27the radius of the smaller circle. So
34:29what is this situation? What is this
34:31situation? That it is a circle within a
34:33circle with two different centers. Okay
34:35? And the smaller circle is completely
34:37inside the circle. Look, I can say that
34:40in this condition, I can say that the
34:43smaller circle is, the smaller circle
34:46is completely, the smaller circle is
34:49completely inside the larger circle,
34:52completely inside, okay, completely
34:55inside, okay. The smaller circle is
34:59completely inside the larger circle.
35:01What is the topic, what is the heading
35:02of the topic? Circle within a circle,
35:05circle within a circle, okay, and and
35:07not touching the circle, and not, not
35:10tangent to the circle. Let's see, this
35:13is, not not touching the circle. That
35:16is, the smaller circle is inside the
35:18larger circle and and not touching the
35:24larger circle. And not touching the
35:26larger circle. Because the next
35:28condition I will tell you in a while
35:29will be completely different. It will
35:31be very similar to this one. It will
35:32just have a slight addition. That's why
35:35I wrote these statements along with it.
35:36Condition number one, can you see it?
35:39The smaller circle is inside the larger
35:41circle. The smaller circle is not
35:42touching the larger one, by the way.
35:44Okay? Now, in this, in this, I will
35:45show you one more condition. Okay? In
35:48this, I can also show you another
35:49situation. And the same condition that
35:52we saw earlier, I can define it in
35:54another way like this too. Now, even if
35:56you just pick this up and do it like
35:59this. I am showing you something like
36:00this. Observe it for just two minutes.
36:02Okay? Come on, here it is. Now I can
36:04show you the same thing like this, no
36:06doubt. Here it is, it's the same thing,
36:07right? You can rotate it, it doesn't
36:10matter, do it like this, it's the same
36:11thing, son. So even now, I would still
36:14say that the distance between the two
36:16centers is, look, the distance between
36:18the two centers plus the radius of the
36:19smaller circle is less than the radius
36:21of the larger circle, less than the
36:23radius of the larger circle. So,
36:25technically this is the same thing,
36:26it's just one thing. Okay, alright,
36:28good. But not touching, the smaller
36:30circle is inside the larger circle but
36:32not touching the larger circle.
36:33Situation number two, now I will show
36:35you another situation. First, you
36:37observe this question, this condition,
36:39this situation. Then we will see what
36:41needs to be done in the question.
36:43According to the question, we will work
36:44accordingly. Situation number two. Now
36:46look, I will show you a new thing, a
36:48new diagram. Here it is. I will plot a
36:50circle again and show you. Here it is.
36:52Now I am showing a new situation. Here
36:54it is. Here it is. Now, let's suppose
36:57this is the center of the larger circle
36:59for you. Okay? We know its center is C1
37:02and its radius is R1 for the larger one
37:03. Now look, I will plot another circle
37:07inside this. Note that I have plotted
37:09another circle inside it right here.
37:11Now, look at the interesting part, what
37:13is new this time? Here you go. What did
37:16I do now? I plotted another circle
37:18inside it. Okay? Or let me do one thing
37:29. I will show it like this. Here it is.
37:35The situation is something like this.
37:37Here it is, now what happened in this
37:39condition? Now the interesting thing is
37:41, the interesting thing is that what I
37:43...okay, this is the smaller circle and
37:45this is the larger, and this is the
37:46smaller. Here it is, okay. Where is the
37:49center of the smaller one? Here is the
37:50center of the smaller one. This is the
37:52smaller's center, the smaller's center.
37:55What is the radius of the smaller one?
37:56The radius of the smaller one is r2.
37:58What is the radius of the larger one?
38:00The radius of the larger one is r1. The
38:02radius of the larger one is r1. Then
38:05you see what happened in this condition
38:08? What is the difference between this
38:10and the previous question? Look closely
38:12. I can also say that the radius of the
38:14larger circle is two times the radius
38:16of the smaller circle. I can say that.
38:19I can also say that the diameter of the
38:21larger circle. I can also say that the
38:25diameter of the larger circle, the
38:27diameter of the larger circle equ...
38:33Okay, this. Wait, let's make it even
38:35easier, let's make it easier. Hold on,
38:37let's do it like this. Radius of the
38:40larger circle, radius of the larger
38:44circle, okay. Radius of the larger
38:47circle. Is equal to the diameter of the
38:52smaller circle, Equ...diameter of the
38:58smaller circle. Getting a clear idea.
39:01Look at the diagram. Definitely getting
39:03a clear idea. What is the radius of the
39:06larger one? The larger's radius has
39:09become the smaller's diameter. Meaning
39:12R1 is two times of R2. And you see what
39:14the condition is in this? Smaller
39:16circle is inside the larger circle and
39:19touching the larger circle. In that one
39:22, the smaller circle was completely
39:23inside. Okay? The situation was the
39:26same. But there was a difference of one
39:28thing. Look, here it is, the situation
39:30was the same but there is one thing
39:33that is different, see I will copy it
39:35here, the situation was almost the same
39:37, okay, but what was in that one was it
39:40wasn't touching, and in this one, the
39:42smaller circle is completely inside the
39:45larger circle and touching and
39:47completely inside the larger circle.
39:49Okay? It is completely inside the
39:51larger circle and it is touching it.
39:55Touching it. Okay? That was just the
39:57difference. There was a slight
39:59difference between the two situations.
40:03Okay? What was the condition in the
40:07first situation? In the first situation
40:11, your smaller circle was not touching
40:13the larger circle. In the second
40:16condition, the smaller circle is
40:19touching it. And what is the situation?
40:22The situation is such that the radius
40:24of the larger one has become the
40:25diameter of the smaller one. It became
40:27the diameter of the smaller one. Is
40:28that clear, kid? It became the diameter
40:30of the smaller one. Done. Well, this
40:32condition can also be like this. Look
40:33at this situation. Now I will show you
40:35another condition in this. Let me show
40:37you one more condition in this. Look at
40:39this. We will look at another condition
40:40in this same problem. Here it is.
40:42Suppose this is your larger circle.
40:43Here is the larger one. Okay? What is
40:46the center of the larger one? The
40:47center of the larger one is C1. And
40:49what is the radius of the larger one?
40:50R1. And similarly, look, I have shown a
40:53small circle. Here it is, this is the
40:55smaller one. What is the center of the
40:57smaller one? C2, and the radius is R2.
40:59Okay? Center of the smaller one is C2
41:02and radius is what? R2, here it is.
41:04This is such a circle. This is the
41:05smaller one. Here it is. Okay? So, I
41:08have created another new situation.
41:09Here it is. Now, if I join these. Here
41:11it is. So, what is the condition here?
41:13What was the center of the larger
41:15circle? The center of the larger circle
41:17was C1. What was the center of the
41:18larger circle? C1. Okay? So, what is
41:22the situation now? The situation now is
41:24this. The situation now is that I can
41:28say the distance between the two
41:31centers plus the radius of the smaller
41:35circle is equal to the radius of the
41:38larger circle. Is equal to the radius.
41:41It is clearly visible. This is the
41:43radius of the larger one. What did it
41:44become equal to? The distance between
41:46the two centers plus the radius of the
41:47smaller circle. Look, what was in that
41:50one? What is the difference between
41:51this and that? Distance between the two
41:54centers plus the radius of the smaller
41:56circle. The radius of the larger one
41:57was greater than that. Here it is equal
41:59. So, what is it here? The smaller
42:01circle is completely inside the larger
42:03circle and touching the larger circle.
42:06The condition is the same. This is also
42:07Case B, by the way. But I have shown
42:10Case B twice. There can be two
42:13situations in Case B. Two situations
42:15are possible in Case B. Look closely.
42:18Two situations are possible. So, I
42:20showed you two or three examples. I am
42:22guiding you forward slowly. Okay? I
42:24showed you two or three examples. Now I
42:26will tell you about one more
42:28possibility, what else could happen.
42:30Okay? Well, we hear another word too.
42:33It is called concentric circles. We
42:36hear another word in math. What is that
42:38word? Look. That word is concentric
42:41circles. Concentric circles. This is
42:49also another condition within circles
42:51inside circles. Another condition
42:53inside circle within circle. Concentric
42:56circles. Now, the word concentric means
43:00circles with the same center. The word
43:03concentric means circles with the same
43:05center. Look at this. We call these
43:09circles with same centers. Okay?
43:14Meaning their centers will be the same.
43:16By the way. The coordinates of the
43:18center will be the same. But the radii
43:21will be different. What will be
43:23different? The radii will be different.
43:24How? Look at this. If I show you a
43:27circle. Let's see. Here it is. Look at
43:29its example carefully. Suppose you have
43:31this circle and you have this circle.
43:33Look what I did. I plotted these two
43:37circles and kept their centers the same
43:39. Here it is. The center of both
43:41circles is the same. Here it is. I kept
43:43the center of both circles the same.
43:45Okay? The larger one is C1. Its radius
43:48is R1. We have written this. Let's name
43:50the larger center C1 and the radius R1.
43:53Let's call it R1. Okay? Now, show R
43:58here. I actually showed R here. If you
44:02want, you can call it R or R1. That
44:04will also work. Then make the smaller
44:07one R2. Then make the smaller one R2.
44:09There is no issue. Is that clear, kid?
44:11We will do either one of these things.
44:14Okay? Look, I will guide you again. If
44:17we have kept this condition, what did
44:19we name the larger one's radius? Let's
44:21show you the larger one's radius so
44:23everything becomes uniform. Named this
44:26one R1 and named this one R2. Okay? So
44:29tell me, what was the condition here?
44:31What was the situation? See, we made
44:33the larger one's radius R1 so that
44:35there is symmetry in everything. So
44:37that there is symmetry in everything.
44:39So look, this entire radius was R1, so
44:42R1 is greater than C1 + C2 + R2. So
44:45what did I do? I just replaced R with
44:47R1 so that everything becomes the same.
44:48It is the same thing. That's it,
44:50brother. What is this entire gap? It is
44:52R1. So, it will be greater than these
44:54two centers plus its radius. Done. It's
44:55finished here. Everything else ahead is
44:58correct. Everything else ahead is
44:59correct. Done. Now look, so I have
45:01ended the confusion of capital R
45:03altogether. Done. Let's look at one
45:06more thing. Now tell me one thing, what
45:08is the center of both circles in this
45:09situation? The center of both circles
45:11is the same, by the way. But the
45:13coordinate radii are different. So,
45:15what is the radius of the larger one?
45:17The radius of the larger one is R1.
45:18What is R1? Obviously, R1 is this
45:21entire gap, right? Who is R1? The
45:23entire gap from here to here is R1. And
45:25what is R2? This small R2. So obviously
45:28, the center of both is the same. The
45:31center of both is the same, but the
45:33larger one's radius is greater. The
45:35larger one's radius is greater. So what
45:37kind of condition is this? This is the
45:38condition of concentric circles.
45:40Circles with the same center but with
45:42different radii. Look, the center of
45:44the circles is the same, but what about
45:46the radius of both? The radius of both
45:48is different. So what is this called in
45:50the language of mathematics? Concentric
45:52circles. Concentric. Okay? So the
45:55smaller one, we had fixed the name of
45:56the smaller one's radius. Who was the
45:58radius of the smaller one? R2. So the
46:01larger one, what was the radius of the
46:03larger one? The radius of the larger
46:04one is this entire thing. The radius of
46:06the larger one is this entire thing. So
46:08obviously R1 will be greater than what?
46:10R2. So the radius of the larger circle
46:12will be what compared to the smaller
46:13radius? It will be greater. So today we
46:16saw four or five situations of this
46:18type. Clear, kid? We saw four or five
46:21situations of this type. Okay? In the
46:24next task, God willing, we will see the
46:27use of all these: disjoint, joint,
46:29intersecting, and circles within
46:32circles. And within the circle within a
46:34circle, these three or four
46:35possibilities can exist from the exam
46:36point of view. Okay, kid? In the next
46:39class, God willing, I will first have
46:41you practice how all of this is applied
46:43. Done, kid. This is also a small point
46:46.

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