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AS Level Mathematics 9709 | Coordinate Geometry 11 | Relation between Two Circles | Raja Shafqat — Transcript

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0:09Assalamu Alaikum beta, where we ended
0:10our last class, that point was called
0:12the 'Relationship Between Two Circles'.
0:15We studied three conditions: disjoint,
0:18intersecting, and tangent circles. Then
0:21we studied 'Circle Within a Circle'.
0:23Within 'Circle Within a Circle', we
0:25also studied whether a smaller circle
0:28is inside, not tangent, tangent, or
0:30concentric circles, which are circles
0:33with the same center. Now come, let me
0:35have you practice some one or two-mark
0:37questions related to that. Look beta,
0:40first check the question, what is the
0:41statement? Part one says to check
0:44whether two circles are intersecting,
0:47disjoint, or tangent to each other. Or
0:50whether one circle is completely inside
0:52the other circle. That is, exactly
0:54those four cases that I explained to
0:56you in the last class. Now we have to
0:59prove those. The interesting thing is
1:02that in all these conditions and cases,
1:04the center and radius of the circles
1:06will already be given. If you look at
1:08your first circle, if I ask you what is
1:10the center of this circle? You will say
1:13three...okay, let me repeat it again.
1:16We already know what the equation of a
1:18circle is, right? The equation of a
1:21circle is always (x-a) ² + (y-b) ² =
1:24r². We know this is the only equation
1:27we have studied so far, and what is the
1:30name of this equation? The name of this
1:33equation is the Standard Equation of a
1:35Circle. This is the only equation you
1:37have studied with me so far. There are
1:39other equations too, but we will
1:41definitely study those later. So, this
1:42is the relationship between two circles
1:44. What are the relationships? We have
1:46to check those now. So tell me about
1:48your first circle—well, I'm not
1:49talking about which is larger or
1:51smaller. How would I know which is
1:53larger or smaller? So for one circle,
1:55its center is 3 and 5, and what is its
1:58radius squared? 16, so then what will
2:00be its radius? Its radius will be four.
2:03Clear beta, its radius has become four.
2:06Done. Come on, its radius is four. Next
2:09, what is the center of the second
2:12circle? -1, 2. Meaning, you simply
2:16match it with this. What is your 'a'?
2:18-1. Match it, what is 'b'? 2, and the
2:21radius squared is 25. So then what will
2:24be the radius? Your radius has become
2:26five. So if I say its radius is r1,
2:28then let's call this radius r2. This is
2:31the situation you have. Now we have to
2:33check what conditions exist here. The
2:35first thing I'll do is show you
2:37graphically so that the concept gets
2:39built up. Look, if you have C1 and C2,
2:41the first thing I did was look at the
2:44gap between the centers of these two
2:46circles. Obviously, I applied the
2:48distance formula. What is the distance
2:51formula? x2-x1 squared. I applied the
2:56distance formula: x2-x1 squared, y2-y1
2:59squared, 2-5, -3 squared, so this comes
3:02to 16 + 9 = 25, and what is the root of
3:0525? The root of 25 is 5. So from this,
3:09I found out what the gap between the
3:11centers of these two circles is. The
3:13gap between the centers of the two
3:15circles is five. The gap between the
3:17centers of these two circles is five.
3:19If you look at the sum of their radii,
3:21what is the sum of their radii? The sum
3:24of the radii is nine. So if any student
3:26remembers the last class lecture, this
3:29is the condition for intersecting
3:31circles. And if any student has a doubt
3:33, I will do one thing. I will also show
3:35you its image so that you remember.
3:38Look here, I will show you its image. I
3:41show an image for everything so that if
3:44any student has forgotten, they can
3:46look at the image and remember. Here,
3:48now take a look at the situation. Okay?
3:52Just like this, by the way, by the way.
3:54Okay? By the way, you can name either
3:57one C1 or C2. I am just showing you a
3:59situation so that you get an idea. Okay
4:02, kid? So that you get an idea. When
4:04you note it down, what is the condition
4:06here? One circle has a radius of four,
4:08and one has a radius of five. Okay?
4:10Let's keep its radius, suppose I am
4:13repeating. This is not an accurate
4:14sketch. By the way. I'm telling you
4:16clearly. Not an accurate sketch. I just
4:18gave you an idea. One's radius is large
4:21. One's radius is small. But one thing
4:23is confirmed. One thing is confirmed,
4:25what is the situation? The situation is
4:27that of intersecting circles.
4:29Intersecting circles. What is the
4:31reason for that? The intersecting
4:32circles situation. And what is the
4:33reason for that? Look. The reason is
4:37that the gap between the two centers is
4:39less, and the sum of the radii is more.
4:41Look here, take this radius as R1. Take
4:43the radius of one as R2. So when I sum
4:46the radii, the sum of the radii will be
4:48greater than the distance between their
4:50centers. Why will it be greater?
4:52Because of overlapping. Because of
4:55overlapping. Look at this. I gave the
4:57same reason when I explained it on the
4:59first day. Because of overlapping.
5:01Because of overlapping. So, this means
5:04the gap between the centers of these
5:06two circles, okay? I am repeating that
5:09this is not an accurate sketch, kid. I
5:12am repeating. I didn't tell you that
5:15this is an accurate sketch. I have only
5:16given you an idea. Clear, kid? Gave an
5:18idea. Obviously, the center of one
5:20circle is different from the other. The
5:22center of one is different. Clear, kid?
5:25And there is a difference in the radius
5:26as well. Obviously, you can name one C1
5:29. You can name one C2. Okay? No issue.
5:32Name them whatever you want. I haven't
5:34sketched an accurate graph. By the way.
5:35Clear? So name the center of one C2.
5:38Name the center of one C1. Done. Okay?
5:41So tell me, this radius here is five.
5:43This radius here is four. So when you
5:46added the radii, this is one radius.
5:49This is the second radius. So when you
5:51summed the radii, an extra portion was
5:53also covered. So that means what type
5:55of question is this? So I should say
5:57this question is about intersecting
6:00circles. What kind of question is this?
6:02Intersecting circles. Intersecting
6:05circles. Look at what the language was.
6:07What do you need to check: whether two
6:09circles intersect, are disjoint,
6:11tangent, or one is completely inside
6:13the other. Okay? So what kind of circle
6:15was this? This was intersecting circles
6:17. Done. This is a rough sketch. By the
6:19way. I have repeated it many times.
6:21Proper sketching would happen only if I
6:23use a compass properly or use some
6:26software. Done. Come on, look at
6:27another example of this type. Okay? So
6:29I have shown you a small example. A
6:31small one, done. Come on, take another
6:34example. Now, do you know we already
6:36know what the standard equation of any
6:39circle is? I will also show you its
6:41sketch after a while. We know, right,
6:44that the standard equation is always (
6:46x-a) ² + (y-b) ² = radius squared. We
6:50read this equation as the standard
6:52equation of a circle. What do we read
6:53this equation as in the language of
6:55math? We call this in the language of
6:57math that, look, this is the standard
6:58equation of a circle. Clear, kid? Its
7:00name is standard equ. And obviously, in
7:03this entire topic, we have taken help
7:05from this equation only. We have taken
7:07help from this very equation. And what
7:08do we read this equation as? In the
7:10language of math? What do we read this
7:12equation as? Standard equation of a
7:13circle. Standard equation of a circle.
7:20Standard equation of a circle. Let's
7:21look at another example. So if you
7:24match it with the standard equation,
7:26let's take a circle with center c1. Its
7:28center is -2 1 and the radius squared
7:32is 64. So what will the radius be? 8
7:34and there is another circle you have,
7:37here is the circle. This is a different
7:39circle. Take the center of this circle
7:42to be 1 -2, here it is. What is its
7:44radius? Its radius is also 8. By chance
7:46, both have the same radius. Clear,
7:49beta? By chance, both have the same
7:51radius. 64 64, by chance, both radii
7:54are the same. Now read the question and
7:56see what your target is. By the way.
7:58Okay? Both have the same radius,
8:00meaning they are circles of the same
8:02size. Now if I tell you, let's name
8:04this R1 and this one R2. So if I ask
8:07you, what is the distance between the
8:08centers of these two? The gap between
8:10the centers is right here. So I applied
8:13the distance formula. So 1-(-2) is 3²,
8:17and -2-1 is -3², so that becomes 9 + 9
8:19, 18. There is nothing logical here.
8:23Applied the Pythagorean theorem, so it
8:26will be 4 point something, clear, beta?
8:30And if I ask you, what is the sum of
8:32the radii? What is the sum of the radii
8:34? Let's apply the distance formula to
8:37these two. Look at the sum of the radii
8:39. 8 + 8 becomes what? 8 + 8 becomes 16.
8:428 + 8 becomes 16. Now tell me, what is
8:45the answer? The answer is 4 point
8:48something. This means that if you have
8:51noted, the gap between the centers of
8:53these two circles is less. Less than
8:56what? Less than the sum of the radii of
8:59both. The sum of the radii is still
9:01greater. So this became a question just
9:02like the previous one. So I will do one
9:04thing. I will also show you its diagram
9:06. Okay? I will show you its diagram so
9:09that you get a clearer idea. Okay? Look
9:15at this. Now if you have any doubt,
9:17look at it this way. Here you go. But
9:20the issue is that by chance, you have
9:24the same radius for both circles. Let's
9:29just give a rough idea of this. Okay? I
9:32am repeating. I never used the word
9:34that I plotted a proper circle. Okay?
9:37Suppose both circles have the same
9:38center. Both circles have the same
9:39radius too. Can you see it? What is the
9:41radius of both? You note it. The radius
9:43of both is 8 and 8. What is the radius
9:45of both? The radius of both is 8 and 8.
9:47Check it. So look, this is eight. This
9:50is eight. So the gap between their
9:54centers is less than the sum of the
9:56radii. So what kind of circles are
9:58these two? This means both of these
10:01circles are also intersecting circles.
10:03So what word should we use? Circles
10:06intersect, or you can say they are
10:08intersecting circles. So what will we
10:10call these circles? We will say what
10:12kind of circles are these? These
10:13circles are intersecting circles. Okay?
10:15Now, let's slowly change the condition
10:17and the situation. So what kind of
10:19circle is this? We can say that, man,
10:21these are intersecting circles. That is
10:23why I am showing this sketch. So you
10:25get a little idea. It is only worth 1
10:27mark. It is a job of one or two minutes
10:29. Alright, let's change the condition
10:32now. Here you go. Look at another
10:33example. We know that we always take
10:35help from the standard equation. And
10:37what is the standard equation?(x-a) ²
10:41+ (y-b) ² = r². Well, so far we have
10:45obviously been doing this continuously.
10:48Tell me the center of the first circle.
10:50Match it with this, so what is a? Match
10:52the 4. -b became 1. So what is b? -1,
10:55there you go. You have its center now.
10:58And if I ask you what is its radius?
11:00Then you will say its radius is 20.
11:02Here you go. Its radius is 20. Check it
11:05again. Its radius is 20. Simple. What
11:08is the square of the radius? 20. So
11:10what is the radius? 20. Now match it
11:13with this. When you match it, tell me
11:15what is its a? a is -3. Meaning-a =
11:19what?-a = 3. So what is a? -3. So the
11:23center of this circle is what? -3 and
11:26match it with this. What is b? Your b
11:28is 5. Here you go, match it. Its b is 5
11:31and its radius is, the square of the
11:33radius is 17, so the radius will be 17.
11:36There you go, nothing very logical so
11:39far. We simply did what we always do;
11:42matched it with the standard equation
11:44to find the center and radius of both.
11:47Well, since I don't know the exact
11:49condition, I checked the gap between
11:51both centers. I applied the distance
11:54formula. Then let's see what the
11:57condition is: (x2-x1) ² and (-1-7) ²
12:00square. Here you go, what else will it
12:03be? 5-1 is 4, 6² is 36, and there's 49
12:07, okay? So that will give you 36 + 49 =
12:1185, there you go, so what answer did
12:12this derive? The answer derived is 85.
12:1785, and by the way, the root of 85 is
12:219.22. 9.22, there it is, we got 9.22.
12:27Now similarly, if you look at the sum
12:29of their radii, what is the sum of the
12:32radii? 20 + 17, here one radius is 20
12:38and one is 17, here you go, 20 + 17. 20
12:45+ 17, so what answer did this derive?
12:488.6. Okay? Okay? The answer derived is
12:528.6. Now you tell me, the distance
12:54between the centers is greater compared
12:57to the sum of the radii. So if any
12:59student remembers, they will
13:01immediately say, "Man, these are
13:02disjoint circles." These are disjoint
13:05circles. Disjoint circles, because the
13:08gap between their centers is larger and
13:10the sum of the radii is smaller. Let me
13:13do one thing. I will show you the
13:15diagram so you have an idea of what
13:16this situation is. This is the
13:19situation we used to call 'disjoint
13:21circles', this very situation. Okay, I
13:24will show you so you can confirm and
13:27match it. Here you go, this is the
13:29situation. Meaning the situation is
13:32like this, look, the situation is like
13:34this. By the way, this is not an
13:37accurate diagram—I have mentioned
13:39this many times—but I am just giving
13:41an idea that the gap between both their
13:43centers was what? 9.22, the gap between
13:45their centers is greater than the sum
13:47of the radii. Look here, the sum of the
13:49radii is smaller. The sum of R1 + R2 is
13:52smaller. So in the distance between the
13:54two centers, this middle portion is
13:56extra. So what kind of circles are
13:57these? We used to call these disjoint
13:59circles. In the two examples you saw
14:01previously, both happened to be
14:03intersecting circles. Clear, kid? So
14:05take another look. This is a one-mark
14:08thing. So what kind of circles were
14:09these? We should call these circles
14:11disjoint circles. Disjoint circles.
14:14Clear, son? Disjoint circles. So these
14:17are just small little concepts. If you
14:19remember the conditions, then it will
14:21be easily managed. Okay? Easily. Only
14:23one or two questions have appeared on
14:25this in past papers to date. So,
14:26definitely, when I cover the past
14:27papers, those things will be repeated.
14:29Take it, son, things will be repeated.
14:31Come on, look at another example. First
14:33, check the question. What is the
14:34statement? Read it. We know, as always,
14:37we first take help from the standard
14:39equation. x-a² y-b² = radius squared,
14:46here you go, = radius squared, done son
14:49. Okay, now you match it with this.
14:51What is this thing? I can also say this
14:52as x-0² So that means if I ask you
14:56what the center of this circle is? Then
14:57you will say 0, and match this with
14:59this, then what is your b? It becomes
15:01two. There you go, here is a, b. This
15:04is your a, b. The radius squared is 10,
15:07so if I ask you what the radius is,
15:09what is its radius, then you will say
15:12root 10, there you go, root 10. Match
15:16this one with it too. When you match
15:18this with this, what is the a of this
15:20circle? The a of this circle is, a is
15:22three, and what is b? b is -3. So its
15:25center is 3, -3. There you go, this is
15:28also 3, -3. There you go, and what is
15:31its radius? The radius squared is 12.
15:33The radius squared is 12. So that means
15:36if I ask you what its radius is? Then
15:38you will say its radius is root 12.
15:41Root 12. The first thing I did was look
15:43at the gap between the centers of these
15:45two circles. The gap between the
15:46centers of these two circles is right
15:47here. If I look at the gap between the
15:49centers of these two circles. 3-0 this
15:51becomes 3, 3-0 3² And -3-2 -5², so
15:57what is the answer to this? 9 + 25, and
16:01what does 9 + 25 mean? 34 9 + 25 means
16:0734, there you go, 34, now let's
16:10simplify this a bit. The answer for
16:13root 34 is 5.83, there you go. What is
16:18the derived answer for this? 5.83, this
16:19is the distance between the two centers
16:22. If I ask you what the sum of the
16:24radii is? If I ask you what the sum of
16:26the radii of these two is? The radii,
16:29so the sum of the radii is root 10 and
16:31root 12, there you go, root 10 and root
16:3312, here it is, so if I look at it
16:35numerically like this: 3.16 + 3.46,
16:37then what is their numerical answer? 6
16:44Uh, 6.63, there you go, 6.63, the sum
16:47of the radii is greater. Greater than
16:50what? The sum of the radii is greater.
16:52Distance between two centers. Yes,
16:55that's fine. Because what is the root
16:58of 36? The root of 36 is 6, right? Is
17:01that clear, son? So what kind of
17:02circles are these? They are telling us
17:04this too. Look again, the distance
17:06between the centers is less. The sum of
17:09the radii is more. So that means this
17:11is an intersecting circle situation.
17:13Look how? I will guide you. This is
17:16also an intersecting circle situation.
17:17Look, like this. What is the condition?
17:24Yes, that's right. I am showing this
17:28situation again so that you can match
17:30it. Alright, son? I'll show it so you
17:33can compare. Okay? This is this
17:39condition. Look. The gap between the
17:44centers is less. Less than what? Look,
17:47the gap between the centers is less
17:48than the sum of the radii. The sum of
17:51the radii. Alright, son? It is less
17:54than the sum of the radii. You can take
17:56another look if you want. There is no
17:58rush, so you can match this situation.
18:00Look, when I checked the sum of the
18:02radii, look, this is R1. This is R2.
18:05One is R1, one is R2. R1 is from here
18:08to here, one. R2 is from here to here,
18:10two. So in R1 and R2, this portion
18:12became overlapping. This portion was
18:14counted twice. So it became overlapping
18:16. So what kind of situation is this?
18:18What shall we call this? Intersecting
18:20circles. What does the word "
18:21intersecting circles" mean? That these
18:24two circles intersect each other. Our
18:26target is not the point of intersection
18:28. We just have to justify or prove it.
18:29Clear, son? Just have to prove it. Done
18:32? I will show you one or two more
18:33examples. Then we will move forward
18:35slowly. Okay? I will show you one or
18:37two more examples. Look at the question
18:39. Well done. First of all, what is the
18:41situation? We know in every question,
18:42what do we compare it with? If you
18:44remember, in every question, the first
18:46thing we do is use the standard
18:48equation. x-a² + y-b² = r², we call
18:53this the standard equation of a circle.
18:57Come on, if you match it with this, a
18:59becomes 0, then it becomes x². If b
19:02becomes 0, it becomes x squared. So it
19:04means that your first circle is the
19:06first circle. What is the center of
19:08this circle? Look, the center of this
19:11first circle is 0 and what is its
19:13radius? The square of this radius is 30
19:16. By the way, the square of the radius
19:18is 30. So if I ask you what its radius
19:20would be? So you would say its radius
19:22will be 30. Its radius will be 30. Is
19:27that clear, son? This is its radius.
19:29Here you go, 30, done, son. Okay, in
19:32the same way, match this with this.
19:34Here you go. What did-a become? It
19:37became one. So then what did a become?
19:38-1. a became -1, and what is its b? One
19:42. So then, what is the center of this
19:44circle? Look, the center of this circle
19:47is -1, 1; -1, 1, there it is, the
19:51radius squared is six. So then what
19:54will the radius be? The radius will be
19:57this; its radius will be 6, here it is.
20:01Its radius will be 6, is that clear,
20:04son? The radius will be 6. Done. Okay,
20:08now here is an interesting point. Let
20:10me do one thing. Let me show you the
20:11answer numerically as well. 30, 30 will
20:14be 5.5, here it is, this is 5.5, and
20:19what is 6? 6 is 2.4. What is its radius
20:25? 2.45, here it is. So its radius is
20:295.5. It is quite large. This is 0. 0, 0
20:32, there it is. Its center, so what is
20:35its center? Let me show you an idea.
20:37Look, I am giving you a rough idea so
20:39that you understand a little bit. Look
20:42son, there is a circle, its center,
20:45suppose a circle like that, its center
20:49is 0. Here it is, suppose this is 0,
20:52this is 0, so this is 1, 2, 3, 4, 5.5.
20:55Suppose it is 5.5 from every end.
20:57Suppose from every end, I am giving a
20:58rough idea. By the way, I am giving a
21:00rough idea. 1, 2, 3, 4, 5.5, if you
21:03want, 1, 2, 3, 4, 5.5, I have just
21:05given an idea by the way. Okay? I am
21:09repeating myself. This is not an
21:10accurate diagram. I am just giving an
21:12idea. Alright, son? I always say this.
21:14Here you have the question. The center
21:16is found. Radius means distance from
21:18the center to the circumference.
21:20Distance from the center to the
21:21circumference. I showed a graph sketch
21:23like this. Alright? This is circle C1.
21:27This is circle C1. Here it is. C1 and
21:31its center was 0 and what was its
21:33radius? 5.5, I am just giving an idea.
21:375.5, now the smaller circle inside,
21:39there is another circle inside, smaller
21:42. Alright? Now let's look at that too.
21:44What kind of circle is that? What is
21:45the smaller circle like? Look, its
21:47center is -1, 1, -1, 1, it will be
21:49somewhere around here. I am giving an
21:51idea like this. It will be somewhere
21:52around here. Obviously, here you go. I
21:54am giving an idea so that you remember.
21:56Alright? Here you go. -1, 1 will be
21:57somewhere around here. Its radius 2.45
22:00will obviously be less than 5.5, right?
22:042.45 would be somewhere around here. It
22:06will definitely be less than 2.5. So
22:07that means your second circle is a
22:10circle of this type. Just giving you an
22:12idea, by the way. Okay? Here, a circle
22:15of this type. You can change the color
22:17if you want. Here, it's a circle of
22:19this type. The situation is something
22:21like this, just gave you an idea, by
22:23the way. Gave an idea. So what is the
22:25situation here? Look, we name this
22:28circle C2, and its radius—you already
22:32know its center—is -1, 1, right here,
22:36and its radius was 2.45; 2.45, meaning
22:39it was less than 5.5. So I can say that
22:43the larger circle contains the smaller
22:45circle. The larger circle, or you can
22:48say, the smaller circle is completely
22:50inside the larger circle and not
22:51touching the circle. Remember anything,
22:54sir? Which condition is this? So I can
22:56say, man, which condition is this? This
22:58is the "circle within a circle"
23:00condition. This is the condition for a
23:02circle within a circle. Circle within a
23:04circle. Meaning I can also say that the
23:06smaller circle is completely inside the
23:08larger circle. I can say the smaller
23:12circle is completely inside the larger
23:15circle. The smaller circle is
23:18completely. Okay? Completely inside the
23:22larger circle; the smaller circle is
23:26completely inside the larger circle,
23:29done, and not touching the larger
23:33circle, not touching the larger circle,
23:37just done, not touching the larger
23:40circle. Okay, child? So we have studied
23:43all these conditions. So, just to be on
23:45the safe side, I showed you a rough
23:46sketch. A rough sketch so you could get
23:48a bit of an idea. Clear, child? Let me
23:51show you one or two more examples. Here
23:53you go. Come on, suppose this is the
23:54question. I have a doubt about this
23:56question, and let's check what the
23:58examiner wants to do this time. Okay?
24:00Well, look, what do we always compare
24:03it with? The x-standard form. x-a²
24:09y-b² = radius squared. Match this with
24:13this. -a became 3. So what did a become
24:15? -3; what did b become? b became 2. So
24:19, what is the center of this first
24:21circle? The center of this circle is -3
24:24, 2; what did the radius squared become
24:27? It became nine. So what did you
24:28consider its radius to be? Matched it
24:31with three. Match it with this one. -a
24:33became 7. So what did a become? What
24:36happened to -7 a? -7, there you go. And
24:39matched it with this. -b became 1. So
24:42what did b become? b became -1. There
24:44you go. This is the center of the
24:45circle. And what is its radius? Its
24:47radius is two. There you go. What did
24:48its radius become? Two. There you go.
24:51Its radius came to two. Done. The
24:53situation is before you. Clear? Now
24:55let's see what I can do with this
24:57question. One radius is three. One
24:59radius is two. Look, if I ask you what
25:01the distance is between the centers of
25:03these two circles? What is the distance
25:05between the centers of these two
25:06circles? Look closely. If I apply the
25:09distance formula here, the distance
25:11formula will be -7, -3, -4 squared. And
25:14this will be -1, -2, -3 squared, 25,
25:17its root is 5, its root is 5, it's
25:19right in front of you, its root is 5,
25:22simple, and if I say what is the sum of
25:25the radii? So when you sum the radii,
25:28what is 2 + 3? 2 + 3 is 5. So the sum
25:32of the radii is exactly the same as the
25:34gap between their centers. The gap
25:37between their centers. So this means
25:39that what condition is this? This is
25:41the tangent circles condition. Let me
25:44do one thing for you. I will show you
25:46its shape so you remember. So what
25:48condition is this? This is the tangent
25:50circles condition. Clear? This is the
25:54tangent circles condition. Okay? I'll
25:56show you so you get some idea. Here,
26:00this is the situation. So what is the
26:03condition here? Here is the tangent
26:05circles condition. There, I'll mark it.
26:10Here the condition is the tangent
26:11circles condition. What are these two
26:13circles? They are tangent to each other
26:15. Okay, child? There you go. These two
26:18circles are tangent to each other.
26:20Because when two circles are tangent,
26:22the sum of their radii is equal to the
26:25distance between their centers. Equal
26:27to the distance between their centers.
26:29So what is proven is that the two
26:30circles are tangent to each other.
26:32Tangent to each other. So what kind of
26:34circles are these? The circles are
26:35tangent to each other. Meaning there is
26:37a one-point intersection between them.
26:38Clear, child? There is a one-point
26:40intersection between them. Done. Let's
26:43try more. I will show you one or two
26:46more examples. Just random. Okay? Come
26:48on, suppose you have this situation.
26:51Okay? We will try one or two more
26:52questions like this. This situation is
26:55x² + y² = 4 and there is another
26:58circle. Suppose x² + y² = 1, there
27:01you go. Let's just give a random
27:03example like this. I need to check what
27:05the relationship is between the two
27:07circles. I need to check what is the
27:09relationship between the two circles? I
27:13don't know if the circles are
27:14intersecting, disjoint, tangent, or a
27:17circle within a circle. Oh, I
27:20remembered that the first thing we
27:21always do is compare it with what? With
27:23the standard form. It's always (x-a) ²
27:27+ (y-b) ² = r², clear kid? = r
27:32squared, done. Now when you compare it
27:36with this, if 'a' becomes zero, 'a'
27:38becomes zero, and 'b' becomes zero,
27:41then it becomes x² + y² = r². So
27:44when you look at the first circle, look
27:46at this circle, this circle, it means
27:48what is the center of this circle? 0,
27:51what is the radius squared? The radius
27:54squared is 4. So that means what is its
27:57radius? Its radius is two. Similarly,
27:59match this one too. When 'a' and 'b'
28:02become 0. Then what is left here? x²
28:04y², we have already studied all this.
28:07Equals to the radius squared. Done. Now
28:10look, if I look at this circle, what is
28:12the condition in this circle? Then you
28:13would say, hey, what is the center of
28:15this circle? Again 0, but what is its
28:18radius? The radius is one. So if you
28:20look at the relationship between these
28:22two, hey, what is the condition? What
28:24is the relationship between the two?
28:25Look, both have the same center by the
28:27way, but the radii are different. The
28:30radius is different. One radius is
28:31larger, one radius is smaller. So that
28:33means which situation is this? We had
28:35studied a condition about concentric
28:37circles. We had studied a condition.
28:39The condition was called concentric
28:41circles. Circles with the same center.
28:43Look at this. One circle has a radius
28:46of two. Meaning, this is the situation.
28:51Okay? I showed you a rough idea of the
28:53condition like this. The condition is
28:55somewhat like this. One circle is
28:58larger. Okay? Its radius is also two.
29:05I'm just giving an idea like this. Okay
29:07? Come on, here it is. Its radius is
29:08two. Meaning from here it is also two.
29:10From here it is also two. From here it
29:11is also two. From here it is also two.
29:12There you go. Alright, this is the
29:14larger one. Then the other circle is
29:16smaller. Its radius is one. Okay? What
29:19is its radius? It is one. Okay? What is
29:22its radius? It is one. There you go.
29:24Now we have plotted that circle too. So
29:28now, observe one thing. What is the
29:30same in both? Both have the same center
29:32. By the way, both circles have the
29:34same center. This gives us an idea that
29:36look, both these circles have the same
29:38center. So I can say that center C1 I
29:42can say center C1 = C2. So, what does
29:46this mean? It means same centers. Same
29:49centers. But the radius is not the same
29:52. Sorry, the radius is not the same, by
29:54the way. So, radius 1 is not equal to
29:56radius 2. They have the same center but
29:59different radii. There are different
30:01radii. Okay? There are different radii.
30:04If the radii are different, it means
30:07that the circles are concentric circles
30:11. What did we call these circles? We
30:13called these circles concentric circles
30:16. So what does the word concentric
30:17circles mean? Circle within a circle.
30:20So what is the center of both? By the
30:22way, I plotted a rough sketch. This is
30:23not an accurate diagram. So what would
30:25I call this? What kind of condition is
30:27this? This is the condition of
30:28concentric circles. Concentric means
30:30circle within a circle. Concentric
30:32means circle within a circle. Clear,
30:35beta? Circle within a circle. Let's try
30:37another example. Look at this. Then I
30:39will show you one or two more examples.
30:41Suppose this is a question: x² + y² =
30:454, this is one circle, and there is
30:48another circle, suppose you have this
30:51circle (x-1) ² + y² = 1, I need to
30:54check the relationship between these
30:58two circles. There must be something, I
31:01don't know. What is the relationship
31:03between the two circles? I don't know.
31:05Well, we know, as always, whose help do
31:07we take first? We always take help from
31:10the standard equation. In this whole
31:12topic, you have studied only one
31:14equation so far. The name of that
31:15equation is the standard equation. The
31:17name of that equation is the standard
31:19equation. Okay? Standard equation of a
31:21circle. The name of that equation is
31:27the standard equation of a circle. Okay
31:29, beta? Well. Now compare it. If a
31:34becomes 0 and b becomes 0, then x²,
31:36y², r²; so it means that this circle
31:39of yours, what will be the center of
31:41this circle? Tell me, 0. What will the
31:45radius be? The radius will be two.
31:47There it is in front of you, and then
31:49after that, compare it with this circle
31:51. Its 'a' is 1, and if 'b' becomes zero
31:54, then this condition will be met. So,
31:56what is the center of this circle? The
31:58center of this circle is 1 0, the
32:00center of this circle is 1 0, and what
32:03is its radius? Its radius is one. Look,
32:06its radius is large. Its radius is
32:08small. Let me give you an idea like
32:10this. Okay? I'll give you an idea so
32:13that you are clear about what happened
32:14this time. Basically, look. Look, what
32:19is the number one situation? I am
32:21giving you an idea. Okay? What is the
32:24center of the large circle? Look, our
32:26first target is to check the
32:28relationship. Here is two. So, what did
32:33I do? I just randomly plotted a circle
32:35like this. Okay? It’s a rough sketch,
32:37by the way. Meaning, this gap is also
32:39two. This gap is also two. This gap is
32:41also two. This gap is also two. No,
32:43wait. Let’s make it a little more
32:52accurate. One, let's make it a little
32:54more accurate. Suppose this is two.
32:56Suppose this is two. Suppose this is
32:58two. Suppose this is two. This gave a
33:00rough idea. Okay? A rough idea. So, I
33:02plotted this circle. What is the center
33:04of the circle? 0, the radius is two.
33:06Done. I wrote the name of this circle
33:09outside. This circle is C1, by the way.
33:11I wrote its name outside to identify
33:13this circle. This circle is your C1.
33:15There you go. Let’s move on. And what
33:17is its radius? By the way, its radius
33:18is R1. And the radius came out to be
33:20two, by the way. The radius came out to
33:21be two, by the way. Clear? Let’s move
33:23forward. Look at the next circle. What
33:25is the center of this circle? By the
33:26way. What is the center of this circle?
33:28The center of this circle is 1 0, it
33:30will be somewhere here. Obviously, 1 0
33:32will be somewhere here. That is how we
33:35plotted it. We said this is two. This
33:37gap is also two. Look, this gap is also
33:39two. Meaning 1 2, 1 2, 1 2, 1 2, okay.
33:44Now, what is the condition for the next
33:46circle? Look. Its center is 1 0, radius
33:49one. 1 0, radius one, meaning its
33:52center is right here. Its center is
33:54here and the radius is one. A radius of
33:57one means the gap from all sides is...
33:59...one each. Meaning, the gap from here
34:01will also be one. The gap from here
34:02will also be one. The gap from here
34:03will also be one. The gap from here
34:04will also be one. So, it means this
34:06circle is something like this. Clear? I
34:09gave you a rough idea like this. So, it
34:12means which condition was this? This
34:15means I can say that the radius of the
34:17larger circle is two times the radius
34:19of the smaller circle. That is, I can
34:22also say that the radius of the larger
34:24circle is equal to the diameter of the
34:26smaller circle. The radius of the
34:29larger circle is equal to the diameter
34:31of the smaller circle. The diameter of
34:35the smaller circle. Meaning, see, this
34:37smaller circle is completely inside the
34:39larger circle. So, what is this
34:41situation? The circle within a circle
34:43situation. The circle within a circle
34:45situation. The smaller circle is inside
34:48the larger circle and is also touching
34:50it. It is also touching it. This is a
34:52tangent, right? Obviously, this
34:54condition itself means a tangent. So,
34:56indirectly, which condition is this?
34:58Indirectly, the condition is the same,
35:00circle within a circle. Indirectly,
35:05which condition is this? Circle within
35:07a circle. So today I showed some small
35:10examples so that whatever you studied
35:12in the last class gets revised. Clear,
35:14kid? So that whatever was studied in
35:16the last class gets revised. Done. Come
35:18, let me show you a good question. Here
35:20it is. Now, please read the question. I
35:22will guide you. Go ahead, read it. Well
35:24done. First, read the language properly
35:26. Two examples have been given to you.
35:28And what is the examiner saying? First,
35:29you have to read the language. Well
35:30done. I will guide you. This requires
35:32slightly better math. Coordinate
35:34geometry will also be used here. So, we
35:36have always read before that what do we
35:38call this equation in the language of
35:40math? We read it in the language of
35:42math as the standard equation of a
35:44circle. We read this equation in the
35:46language of math as the standard
35:47equation of a circle. Okay? We have
35:49already studied this. The name of this
35:52equation is the standard equation of a
35:53circle. Come on, here it is. Now match
35:55this with the first one. If 'a' becomes
35:57zero and 'b' becomes zero, then what
35:58does this equation become? x², y² =
36:00r², is it like that, kid? x², y²,
36:04r²; so if I ask you what is the center
36:06of this circle? Then you will say 0,
36:09and what is its radius? The square of
36:12the radius is 25. So we will say the
36:13radius is five. That's the end of it.
36:15This is a small circle; we have seen
36:17what the center and radius of the first
36:19circle are. Done, kid. Oh, we are stuck
36:22here. We are stuck. Now I remember that
36:27, well, we studied completing the
36:29square in the first class. Okay, kid?
36:32Now I am taking a little help from that
36:33. Later, you will also study a new
36:35equation that we haven't studied yet.
36:37By the way. Here is the question in
36:39this shape. Now, I’m stuck here. I
36:41got confused because the question is
36:44not in the ideal form that we normally
36:46have. So, what do I need to apply now?
36:49I need to apply completing the square.
36:52I need to apply completing the square.
36:55Done, son. This needs to be applied.
36:57What did we learn in completing the
36:59square? First, separate the x terms.
37:02Here, we have separated the x terms. We
37:05have already studied this in completing
37:06the square. Then we learned to separate
37:08the y terms. Because obviously, first
37:10there is the square of the x term. Then
37:12there is the square of the y term. What
37:14will you do in the next step? Here, you
37:16combined the y terms. y²-18y and 125
37:20is copied as it is. Now my completing
37:24the square must be perfect. Okay, son?
37:27Now the completing the square must be
37:29perfect. Now apply completing the
37:31square to this. What happened here? a²
37:33-2*a*b, we know how b is found. It's
37:37always coefficient of x/2, here it is.
37:4012 squared is added. 12 squared is also
37:43subtracted. Here it is. 12 squared is
37:46also subtracted. Now complete this
37:48formula too. What will this become? y²
37:50, there is a minus in the center, so
37:51the formula will also be for minus. 2*a
37:54*b, 18/2 becomes 9. So what did we use
37:58to do? We said 9 squared will also be
38:00added. 9 squared will also be
38:02subtracted and 125 copied as it is.
38:05Here, 125 is copied as it is. We have
38:08obviously done this work already. Now
38:10what will this whole thing become when
38:12completed? Look, this whole thing will
38:14be completed for you. Through
38:17completing the square. In the same way,
38:18this whole thing will also be completed
38:20for you. Through completing the square.
38:23Clear, son? Now let's finally complete
38:25this. So what will this become? This is
38:28x-12 squared and what is 12 squared?
38:31-144, here it is. This is completely
38:34finished. And this remaining was -144.
38:37What did this whole thing become? This
38:39whole thing became y-9 squared and 9
38:42squared is 81 and 125 copied as it is.
38:46Here, 125 copied as it is. Make this a
38:49little easier. This became (x-12)
38:51squared. Make this easier, (y-9)
38:55squared. Let's make this easier, this
38:58will be uh, when simplified 144 +
39:00125-81 this becomes 100, shifted there
39:03it becomes 100, there you go, now the
39:06question is converted to an ideal form.
39:10Now when we match this, a is 1, and
39:12when we match this, b is 9. So from
39:15here I found that the center of this
39:17circle is (12, 9). What is the square
39:20of the radius? The square of the radius
39:23is 100, so you have the radius, sorry,
39:24the square of the radius was 100, so
39:26the radius is 10. This is the center of
39:28the circle. A little extra effort was
39:30required because you had to apply
39:32completing the square. You had to apply
39:35completing the square. That is the only
39:36difference. Alright, so we have done
39:39something or other. Next point. Now
39:41what shall we do? Now tell me one thing
39:44. If I look at the gap between the
39:46centers of these two. Actually, I
39:48applied the distance formula. Here it
39:50is, so this will be the distance
39:52formula. Applied it, look at the gap
39:54between the centers of these two, in
39:55fact, what is written here: "Show that
39:57circles touch each other". Let's check.
39:59So this will be 12 squared and 9-0, 9
40:02squared. 144 + 81 is 225, the answer is
40:0715, the answer is 15, here you go, and
40:10if I sum their radii, if I sum their
40:13radii, if I sum the radii of these two,
40:16then its radius is five. Its radius is
40:1910. So look, it is proved. Yes, that
40:22the distance between their centers is
40:24exactly the sum of their radii. Exactly
40:28the sum of their radii. Look at it
40:31carefully, kid. So basically, the
40:33situation was something like this.
40:36Basically, the situation was something
40:38like this. Such a condition, the
40:42situation was like this. Your smaller
40:44circle, what was the center of the
40:46smaller circle? 0, this was your
40:49smaller circle. What was its center?
40:51Its center was 0. Its radius was five.
40:56Its radius was five. And your larger
40:59circle, your larger circle. Like that,
41:06your larger circle, what was the center
41:09of the larger circle? Anyway, this is a
41:14graph sketch. Okay. Okay, now look,
41:16number one, what is the center of the
41:18larger one? The center of the larger
41:20one was this for you. Here is the
41:21center of the larger one, I will mark
41:24it. The center of the larger one was uh
41:270, the center of the larger one was 12,
41:309 and what is its radius? Okay? Meaning
41:32the situation is not like this. Ideally
41:34, the situation should be like this.
41:36Let me show you by rotating it this way
41:38once. The situation is something like
41:40this if you want to work a bit more
41:41ideally. There it is, meaning your
41:44smaller circle, this is your smaller
41:47circle. Here you go, the smaller
41:49circle's center was 0, what was its
41:51radius? Its radius was 5, what was the
41:53larger circle's center? The larger
41:56one's center was 12, 9 and what was its
41:58radius? Its radius was 10. There you go
42:00. Its radius was 10 and this radius was
42:03r1, here it is. So now tell me, what
42:05was the gap between their centers? 15,
42:08what was the sum of their radii? 15, so
42:11that is what had to be shown. That is
42:13exactly what had to be shown. So we
42:14first, we studied in the last class. We
42:18studied in the last class that whenever
42:20two circles are tangent to each other,
42:22the distance between their centers will
42:24be equal to the sum of the radii. To
42:27the sum of the radii. So this is shown.
42:31So I have written "shown" here. This
42:33was what had to be shown, right son?
42:35First we justified it, it is shown. Now
42:37a little bit of common sense. Okay? Now
42:39you read ahead. I will guide you. The
42:41next part, this is interesting. Find
42:43the coordinates of the point where they
42:45touch. Where they touch. So tell me, by
42:47the way, what is the ratio between them
42:49? 1: 2 If I tell you that, hey, this
42:53point is your target, right? Point
42:55where they touch. So what will be the
42:57name of this point? The name of this
42:59point will be the point of tangency.
43:00Here it is. This point, the one I am
43:03highlighting. Here it is. This point,
43:05let me highlight it in white. Here it
43:07is, this point, the name of this point
43:10should be the point of tangency. So the
43:12examiner is asking you to find this
43:14point. Who is this point? This is the
43:16point where both circles are touching
43:18each other. And this is called the
43:21point of tangency. What is it called?
43:25In the language of math? In the
43:27language of math, it is called the
43:29point of tangency. Where two graphs, or
43:31two circles, are tangent to each other.
43:34The last and final condition. Now look,
43:37I am showing you this same diagram to
43:39explain it to you. Okay? I am just
43:41showing you this same diagram so that
43:42this question is finally completed. You
43:44tell me, you already have this point.
43:46This point is already there. And
43:47somewhere here, both are touching each
43:50other. Somewhere here, both of them are
43:52touching each other. Okay? Here is your
43:55point. They are touching each other
43:58right here. Here, let me mark it in
44:00white. Both are touching each other
44:01here. This is your target. The point of
44:03tangency is your target. Tell me, what
44:06is the ratio between them? The ratio is
44:081: 2. Let me explain it again. This
44:10length is five. This is ten. So, the
44:12ratio is 1: 2. The ratio is 1: 2. So,
44:15on the first day, in the first class,
44:17we studied a formula. We need to find
44:19the ratio. Okay? So, this point is 12,
44:219; the topic was "Finding an unknown
44:24point using ratio." Finding an unknown
44:26point. So, this point of yours is
44:27unknown. Let's name this point P. I
44:30have named this point of tangency Point
44:33P. We studied this on the first day. I
44:36will repeat it again. Can you tell me
44:38the horizontal gap between these two?
44:40And what is the vertical gap? So, tell
44:42me, how much is this gap from 0 to 12?
44:4312. And what is the vertical gap
44:46between them? The vertical gap is nine.
44:48We have already studied this. The name
44:50of the concept was "Ratio." Finding an
44:52unknown point using ratio. Now, can I
44:54say that this point here is your C1?
44:57This point is your C2, and this point
45:00is your P. So, can I say that C1P is
45:02equal to what? 1/3 of the whole
45:04distance. Can I say that, son? Look
45:07again. What is this? The total ratio is
45:09three. So, what did this become? It
45:11became 1/3 of the whole distance. Of
45:13the whole C1C2. Clear, son? Of the
45:16whole C1C2. So, if I ask you what the X
45:18gap is between these two points?
45:20Meaning, my target is to find what this
45:22gap is. This is the X gap and this is
45:23the Y gap. We studied on the first day
45:26that if I need its X gap. If I need its
45:29X difference. If I need the X
45:31difference, what will we say? We will
45:32say 1/3 of the whole gap. What is 1/3
45:35of the whole gap? It is four. And if I
45:37tell you I need its Y difference.
45:39Meaning, if this whole thing is nine,
45:40then what will this be? This vertical
45:42gap. I don't know this. So, what else
45:44can I call it? I can say that 1/3 of
45:47the whole vertical gap is three. So,
45:49what did I finally find out? In the
45:51last step, I finally figured out what
45:55my target was. My point was obviously 0
45:58, right here. So I found out that my
46:00x-difference is four. And this
46:02difference, what is it? It is three.
46:04What is this difference? It is three.
46:06There it is. If I have to move from
46:09here to here. To move from here to here
46:12, I need to move four units forward and
46:15three units upward. Then finally, I
46:18will be able to find this point. What
46:21did we name that point? Point of
46:23tangency. So, how will you find the
46:26point of tangency? Move four units to
46:28the right from zero. That makes it four
46:31. And move three units up from zero.
46:33That makes it three. So that means what
46:35are the coordinates of point P? The
46:38coordinates of point P are 4, 3, which
46:41is your final target. So today we
46:45looked at only one thing. What was the
46:48name of that small point? We practiced
46:50the relationship between two circles.
46:52Meaning, all the possible conditions
46:54for two circles were revised by us
46:57today. We just practiced that. Is it
46:59clear, kid? In the next class, God
47:01willing, we will continue from here.
47:03All right, kid? Goodbye.

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