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Problem 84 Hard Difficulty

The figure shows a sector of a circle with central angle $ \theta $. Let $ A(\theta) $ be the area of the segment between the chord $ PR $ and the arc $ PR $. Let $ B(\theta) $ be the area of the triangle $ PQR $. Find the $ lim_{\theta\to 0^+} A(\theta)/B(\theta) $.

Answer

$\frac{1}{3}$

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Video Transcript

in this problem, it is given that the figure shows a sector of a circle with a central angle tita, right? So we can say this is a sector, right? That is our and this is our, that is radius and this is same pio. And oh and is saying, right, so now we can see here it is given let a heater with the area of this segment B. R. And on pr right? So this this a theater is made up of the line segment Pr and our fear. So let the theater be the area of the triangle. P Q R. S. O P Q R has the area of a triangle, right? Recipe. I suppose this is denoted by beating to right. So we need to find the limit. Pita tends to zero plus eight heater by beating. So what we will do, we will just do one thing. No, we will take the area of this sector. Like you can say this, our pr is included and P O r triangle is included. This whole area. We will subject this from this whole area. Even subject the area of B. Or you're trying to. Right? So how we will do so we need to find out the area first separately for every every single um triangle, right? As well as this arc one? And this triangle. Right? So how we will do so let us see. So we know on the steps, right? What we have to do. So that has assumed the ladies of the sector is oh are all you can say Opie. So is it same? Yes, because this this is the point of this sector. So we can say this is O P E Z equals two R. S equals to us. That is radius of the circle. Now from the figure pig you from difficult. We can say PQ is what? Because nothing but this is this is the bi component of this angle. Right? So we can say that walkie signed peter because it was too are presented. That is O. P. Is are we have a zoom book? So this will give us our scientific or pick you I think now but Cuban you we will just see that it will be oh because tita that is are forced to. All right. So now the area of the sector B O R. Area of this sector B O R. When we what This will be what you are. Mhm, mhm Okay mm Okay, so this will be any idea of this sector. I'm sorry yes, area of the sector P You are, it will be what this will be. This will be given by what uh we know the area of this circle is pirates choir but for 40 to this is sector uh this is a sector 40 to so for every three to divided by two x 2 by S. D. To buy is the whole You can say 360° right for a circle, hole circle. So we we just define it uh sector for a circle it is fire square. For a sector. It should be a petabyte to buy in to buy a square. So you need to remember this. This is very important. Now what we will do we will get half pita R squared as the area of this setup you are. That is as to to this is the representation only as tita. Okay the area of triangle P. O. Q. Is. Now we will find this. After that. We will just subtract from this whole uh area to this area. So we can get the pita and rita. Right? So the area of the china boko is CCTA is equal to happen to base into high. So for triangle P. O. Q. The base bases Oq. And hide as you can say thank you. This is already we have I found out that this value or Q. And P. Q. In terms of course are posted in our scientific to. Right? So now this will give us when by four R squared. Sine theta. Why? Because we know that formula called scientists. That is true scientific data. So now the area of a triangle P. O. R. Is D teeter. So the teeter what is equal to happen to base into height again. So here your base is what? And hi it is PQ. Right? So here we will get half in two hours into our scientific this will give us the area half into our square scientist to. Now for Bt to we will just subtract btw will just subject this whole this whole triangle right? This P O R triangle and peak. We can just subtract this. We will get beat peter. Right? So here beat it. Haven't given by data minus c theater. Right? So we have found above all the values for the theater as well as see theater. So we will get after solving half are square to Scientific a minus sign. Now for 80 to we will just just subtract what we will just take the whole area with this sector. Right? And we will just subtract the area of this whole triangle. That is bu this is very simple. Yes. So this will give us half pita r squared minus half art squared Scientific. Right? So this will give us half hours quiet peek a minus site see So we will find the limit here. Right? So let us let us divide a theater by bt to we need to find out we will just find the limit here. Right? So tita tends to zero. So now what we can see here. Well then we will put values here we will get to know dad. This this you can say this This makes zero x 0 form isn't it? How let us let us solve first this is true. T to minus scientific divided by to scientific. My nurse. Scientific if we put theater is equal to zero in both the numerator and denominator. It becomes zero x 0 form so far zero x 0 form. El hospital rule and hospital rules. Is that lady it? So what do we do in early hospital rule? If we get any form? Like in limits we get a 0x0 form or infinity by infinitive form. We will just differentiated and we will just get the value. So this becomes very easy to solve the limit. So for this we will just differentiated then what leverage The Garden's 2080 to buy BT to right. And this will be limited feed that tends to zero. We will just differentiate numerator as well as denominator. So this will give us to one minus course he tell you divided by we will just differentiate this also we will get to cost minus two was It's because Okay, so now what we are getting now after applying in hospital rules we will just see here whether it is Whether we are able to solve the limits with putting the value zero. Right? So we can see again this this form is becoming again this farm is becoming zero x 0 form again. zero x 0 form. Therefore we will apply. Yeah El hospital rule again. Right, This is very simple whenever how many times it may be But if 0x0, former infinity by infinitive home guns then it should be we have to apply and hospital. Okay so now here we will see this value will come out to be limited, tita tends to zero again. So we will just differentiate it after differentiation. We will get Yeah. Okay. Yeah this is a plus to sign duck divided by this is true. Yeah minus sign peter minus. Mhm. This will be minus minus will be plus. So this is this will cut off each other. We will be getting your award. We will be getting limit. Peter Tends to zero again. This is scientific to yeah society theater upon. Mm Yeah minus sign Pekka this will be sign Peter. I'm sorry this is sign to theater right? This is they should be signed to 10 years. Right? So here it will be two scientists to because Zika so this will be forced to introduce will report here. Okay so we will get here limit. He tends to zero. This will be again the scientists will be taken as a common so we will get shine upon minus one plus 44 speakers. So what we will get here We will just put a diet was 20 again. So if Kita equals to zero then it weakens one by minus one place four course zero. This really big one bite today isn't. And so we can easily say that they was acquired value is limit. He tends to zero 80 to by B. T. To. It was too one x 3. So this is our answer. This was very simple question. I hope you understand the concept. Thanks for watching.