V : jee main /advance class 12 mathematics : AOD class-3( 29 may 2026 )

追加: 2026-05-30

[00:00:24]Good morning.

[00:00:31]Did you attempt this one?

[00:00:34]You do it at home if you have not done it. Not a problem.

[00:00:39]uh because I am planning to do different type of problem in this AOD and then we'll start new chapter.

[00:00:49]>> Okay. One rate and measure problem and one other kind that one Try this one first.

[00:02:36]Take this discussion.

[00:06:46]Do you know now what's in the equation of how to find out the slope of tangent, slope of normal and all those things?

[00:06:54]>> Yes.

[00:08:05]Sir, can I show this one?

[00:08:15]what they have given the equation of normal to the curve the given equation uh which was 3 b + a = -2 by replacing the >> you got 3 b + >> a = 2 >> 3 b + a = -2 how how did you get it >> the first equation equation which is given y = x - a by x + b into x - 2 replace the x and y with 1 - 3 then you get the first equation 3 b + a = -2 >> why a you are saying because this point is lying on this it should satisfy it >> correct >> and the next line is given x - 4 y= 13 >> we can find the slope of the slope and tangent using this uh you have to differentiate this curve dy by dx and then find d minus dx by dy and then equate with it at this point um correct only let me solve it the first part you have done is correct because you have sub because 1us 3 will lie on the curve it should satisfy it you are getting this point second thing is uh finding the doing the dy by dx so dy by dx X B X + B into X - 2 - X - A into 2X + B - 2 uh divided by X + B² into X - 2.

[00:10:07]Now the low the equation of normal at the point is this. So that means slope of the normal given is 1 by 4. Slope of tangent would be -4. This should be equal to dy by dx.

[00:10:30]But we have to find dy dx at 1 - 3.

[00:10:34]So at 1 - 3 this dy by dx would become 1 + b into - 1 - 1 - a into b by 1 + b² into 1.

[00:11:03]So it will become -1 - b - b + a / 1 + b 4² is equal to 1x 4.

[00:11:20]So 1 + b² is equal to 4 a - 4 - 8 b Now let's replace a try to get the values of b only. B² + 2 b + 1 is equal to - 8 b - 4 uh + 4 a would be - 2 - 3 b.

[00:11:54]Let us try to simplify this.

[00:11:59]B ² + 2 B + 1 = - 8 B this uh - 4 - 8 - 12 B ha 20 is 22 B ² + 22 B is 13.

[00:12:24]What is this?

[00:12:51]Oh once I have found out dy by dx it was equal to minus4 now why did I do four this one put it 1x4 >> oh >> I found because this was the value of dy by dx and We actually calculate dy by dx = -4.

[00:13:17]But why did I put 1x4? That was a very silly mistake I did. It's equal to -.

[00:13:23]Now we'll get the right equation. That will be -4 uh - 4 b² - 8 b = 1 - - 2 b + a.

[00:13:38]What the value of a? We had - 2 - 3 V. Let's put here - 1 - 2 V - 2 - 3 B.

[00:13:59]Let's simplify it. Let's take the left part to the right side.

[00:14:05]4 B² that's 3 B 4 B² 3 4 + 1 this three this will come here uh this equation we are Are you getting anything?

[00:15:09]Yes, it I've made too many mistakes.

[00:15:27]too many things.

[00:15:40]Press A.

[00:15:44]Now it'll be this.

[00:16:09]Now it will become b² factor b² + 4 + 3 = 0 b² + repeat you get b + 1 into b + either b = -1 or b = minus 3 then if b = -1 a would be equal to - 2 - 3 - 2 + 3 1. So this is this a is 7.

[00:17:10]If we take this >> but if b was minus one >> then the dator would be zero.

[00:17:17]B is minus one then >> d will be zero in dy by dx value of x is 1 and b is minus one so - 1 z in the value >> right right right it cannot be it's not possible actually so the right set is this a 7 b minus 3 >> yes sir >> put observation so here a plus b they're asking a + b means it is 7 - 3 4. So a + b would be equal to right. It was easy now. It was not difficult actually. I think you also did silly mistake like me.

[00:18:04]>> Yes sir.

[00:18:16]Now pay The answer is four.

[00:23:58]Answer is four.

[00:24:01]>> Yes, sir.

[00:24:07]>> What are the points you're getting?

[00:24:14]Just say the number of points the points the value of the points is >> dot uh - 1 by 3 - 2x3 one and two >> uh one and two correct these are the four points very good.

[00:24:42]Now let me give you one question of uh rate and measures.

[00:25:13]This Right. This one 13.

[00:35:01]>> See, see what is given in the question.

[00:35:08]A wire of length 20 m is to be cut into two pieces. Suppose this is a wire H.

[00:35:14]This is a wire. Its total length is given 20 m. A piece of length L1.

[00:35:19]Suppose this is the piece of length L1.

[00:35:26]Now this L1 is bent to make a square of area L1 area A1. Meaning the total square which is made obviously you know the side of the square is same.

[00:35:42]So that means 4 L should be equal to L1.

[00:35:49]The length of the square, the length of one side of a square will be equal to L1 by 4. Correct?

[00:35:56]So what would be the area of this?

[00:36:01]A1 is equal to L1X4².

[00:36:06]Right? Okay.

[00:36:09]here up to here.

[00:36:13]Now the length L2 and the other piece of length L2 is made into a circle of area A2. This length L2 is bent to make the circle.

[00:36:24]Now if we take its radius the radius of the circle is R. So that means 2 pi r should be equal to l2.

[00:36:34]The length of the circumference should be equal to l2. So r is equal to l2x2 pi. That means the area of this circle a2 should be equal to<unk> into r². That means p<unk> into l2x2 pi².

[00:36:54]It means you will have L2² by 4 pi. I have cancelled p<unk> p<unk> this was p<unk>² pi I have cancelled them.

[00:37:06]This is a2. Now what he's saying he's saying 2 a1 + 3 a2 is minimum.

[00:37:12]So 2 a1 + 3 a2 if we calculate this should be equal to 2 l1² by 16 + 3 l2² by 4 pi this value suppose it's s this is minimum that means d s but you have to get this the problem is you have to get this value in one variable. How you'll do it? You already know L1 + L2 is equal to contains there.

[00:37:51]So you try to make this as in the form of one variable S is equal to L1² by 8 plus 3 L2² by 4.

[00:38:02]Which variable you want to remove L2 or L1?

[00:38:11]It's up to you.

[00:38:13]>> Anyone anyone now? So let's do let's write L1 in terms of L2. So it will become 20 - L1 20 - L2² by 8 + 3 L2² by 4 pi.

[00:38:31]Now this has to be minimum. It's given the value of 3 2 a1 + 3 a2 is minimum.

[00:38:39]So d s by d l2 should be equal to zero is the condition for maximum minimum.

[00:38:46]So it will become 2 into 20 - L2 / 8 it will become -1 + 6 L2 by 4 by uh this zero. So these values is equal to zero. So what you will get 6 L2 by 4 by is equal to 20 - L2 by 4 4 will get cancelled 6x<unk> + 1 into L2 is equal to 20 you have to put the value here X +<unk> into L2 is = 20<unk>. So the value of L2 is 20<unk> by 6 +<unk>. Can you calculate and give me the value?

[00:39:45]L1 is equal to 20 - L2 it will be 120 by 6<unk> + 5.

[00:40:02]This is the value of L2 and this is the value of L1.

[00:40:06]Let me see. Wait what they have asked.

[00:40:10]They want I think I L1 by L2. Now the ratio no need to calculate.

[00:40:18]They want 5 L1 L2 Six 1 by O Six X is the answer.

[00:40:41]Easy.

[00:40:45]Where you got stuck?

[00:40:54]didn't know what to do exactly. Actually you didn't know where to see this kind of this is one one kind of problem that they repeatedly ask actually uh this the differentiation one tangent and normal that type and one more is the the increasing and decreasing type in this what you have to do the one thing that they have asked to minimize or maximize that value you have to make it into one variable like because see the thing I was getting is 2 a1 + 3 a2 I was getting as 2 l1 square by 16 + 3 l2 square by 4 pi now to minimize it I cannot differentiate it because with respect to what variable I'll differentiate there should be at least there should be at most one variable only it can be constant also but it should be one variable only then only I can differentiate with respect to that and then it will become zero and then it will become we can maximize it or minimize it. So that's why I brought into one variable. So in short in these kinds of question you have to find a condition then bring it into one variable then differentiate it make it equal zero.

[00:42:14]Okay let me find one more question for you. Similar one let's try to do this one. This is tangent.

[00:42:47]I'll find one maximum minimum problem written by this problem. Then send put it here. But for now do tangent and normal do That's right.

[00:49:49]Then my Wi-Fi went off.

[00:49:51]>> Your Wi-Fi went off.

[00:49:56]>> It's just disconnected >> now. How you are doing it now?

[00:50:02]>> Hot spot. Uh only 3 minutes are left. So one thing you can do and no need to waste you do one thing. I have see I have selected two questions this one and this one.

[00:50:19]>> Yes sir.

[00:50:20]>> Uh this is tangent and normal and this is of area under the curve. H sorry rate and measure this tangent and normal and this is rates and measure that I was talking about to select I will select one question for you do this and then from the next class we will start a new topic for you okay sir bye-bye sir