Question

A vertical pole of height h stands at the centre O of a circle and subtends an angle $\alpha$ at a point A outside the circle. The circle subtends an angle 2$\theta$ at A. AT and AT' are tangents from A to the circle, then

Answer 1

In $△TA{T}^{{}^{\prime }}$ $\angle OA{T}^{{}^{\prime }}=\theta$
Now in $△POAOA=OP\cot \alpha =h\cot \alpha$ ...(1)
and $AP=OA\csc \alpha$
And in $△OA{T}^{{}^{\prime }}O{T}^{{}^{\prime }}=OA\sin \theta =h\cot \alpha \sin \theta$ (using (1))
and $A{T}^{{}^{\prime }}=OA\cos \theta =h\cot \alpha \cos \theta$
Therefore
Radius of the circle $O{T}^{{}^{\prime }}=h\cot \alpha \sin \theta$
Length of tangent $A{T}^{{}^{\prime }}=h\cot \alpha \cos \theta$
Distance of A from center $O{T}^{{}^{\prime }}=h\cot \alpha \sin \theta$
Distance of A from top of pole $AP=h\csc \alpha$