 Hello and welcome to the session. In this session we discussed the following question which says around balloon of radius r, subtends an angle theta at the eye of the observer while the angle of elevation of its center is phi, find the height of the center of the balloon. Let's see how we find the height of the center of the balloon. Let us take this balloon with center O, then OPOQ are the eye of the balloon which is equal to r. We take this xy to give the horizontal ground and this x is the point of observation. Point x we draw tangents xp and xq to the circle. Next we draw om perpendicular to horizontal ground that is xy. By the question we have that the balloon subtends an angle theta at the eye of the observer. So x is the point of observation. So this means this angle is theta that is angle pxq is equal to theta. Now angle pxO is equal to angle Oxq which is equal to theta upon 2. That is this angle is theta upon 2 and this angle is also theta upon 2. Now angle omx is equal to 90 degrees since we have drawn perpendicular om to xy. In the question we have that the angle of elevation of the center of the balloon is phi. So this means this angle is equal to phi that is the angle Oxm is equal to phi. Now this angle is also of 90 degrees and this angle is also of 90 degrees that is angle Opx is equal to 90 degrees and also angle Oqx is equal to 90 degrees. Now the height of the center of the balloon is equal to om and we have to find this om. So for this first of all consider the right triangle Opx in this we have cos theta upon 2 is equal to hypotenuse that is Ox upon perpendicular which is Op. So this means we get cos theta upon 2 is equal to Ox upon Op which is the radius of the circle that is r. So this means we get Ox is equal to r cos theta by 2. We take this as equation 1. Next consider the right triangle Oxm in this sin phi is equal to perpendicular which is om upon the hypotenuse which is Ox. So from here we get om is equal to Ox into sin phi. Now we have Ox is equal to r cos theta by 2. So this means om is equal to r cos theta by 2 into sin phi that is using equation 1. So this means we have om is equal to r sin phi into cos theta by 2. Therefore we get the height of the center of the balloon from the ground is equal to r sin phi into cos theta by 2. So this is our final answer. This completes the session. Hope you have understood the solution of this question.