00:01
In this problem, it is given that the figure shows a sector of a circle with a central angle theta.
00:10
Right.
00:10
So we can say this is a sector, right, that is r.
00:14
And this is r that is radius and this is same.
00:17
P -o and o -r is same.
00:23
Right.
00:24
So now we can say here it is given let a -theta be the area of the segment p -r and arc.
00:37
Pr, right? so this, this a theta is made up of the line segment pr and arc pr.
00:44
So let b theta be the area of the triangle.
00:51
Pqr.
00:52
So pqr has the area of triangle, right? let us suppose this is denoted by b theta.
01:00
Right.
01:01
So we need to find the limit theta tends to 0 plus a theta by b theta.
01:07
So what we will do? we will just do.
01:12
One thing that we will take the area of this sector like you can say this r pr is included and p o r triangle is included this whole area we will subtract this from this whole area we will subtract the area of p o q triangle right so how we will do so we need to find out the area first separately for every single triangle, right, as well as this arc one and this triangle, right? so how we will do so? let us see.
01:55
So we know the steps, right, what we have to do.
01:58
So let us assume the radius of the sector is or, or you can say op.
02:04
So is it same? yes, because this is the point of this sector.
02:08
So we can say this is opi is equal to or is equal.
02:12
To r that is radius of the circle now from the figure pq from the figure we can say pq is what pq is nothing but this is this is the y component of this angle right so we can say that op sine theta pq is equal to opi sine theta that is op is r we have assumed above so this will give us r sine theta for pq right now for for oq value we will just see that it will be o p cos theta that is r kosa.
02:50
All right.
02:51
So now the area of the sector por, area of this sector por will be what? this will be what? por.
03:12
So this will be area of this sector.
03:18
I'm sorry.
03:24
Yes, area of the sector por.
03:25
It will be what this will be this will be given by what we know the area of the circle is piar square but for for theta this is sector uh this is a sector for theta so for every theta divided by two pi two pi is the two pi is the whole uh you can say a 360 degree right for a circle whole so we we just define it uh a section for a circle, it is piar square.
04:05
For a sector, it should be theta by 2 pi into pi r square.
04:09
So you need to remember this.
04:11
This is very important.
04:14
Now, what we will do? we will get half theta r square as the area of the sector por, that is s -theta.
04:26
This is representation only as theta.
04:29
Okay.
04:29
The area of triangle p .o .q.
04:35
Now we will find this.
04:37
After that, we will just subtract from this whole area to this area.
04:44
So we can get the a theta and b theta.
04:47
Right...