Question

Let α1 be a circle with centre O and AB be diameter. Let P be a point on OB different from O. Suppose another circle α2 with centre P lies in the interior of α1. Tangents are drawn from A and B to the circle α2 intersecting α1 at A1 and B1 respectively such that A1 and B1 are on opposite side of AB. If A1B = 6, AB1 = 18 and OP = 10, find radius of α1.

Answer 1

$\sf\large{\underline{\underline{ Given:}}}$

$\sf\large{\underline{\underline{ In\:Triangle\:APQ, \:We \:have, }}}$

$\sf\large{\red{\implies sin\theta = \dfrac{y}{x} + 10 \: \: .......(i)}}$

$\sf\large{ \: \: }$

$\sf\large{\underline{\underline{ In\:Triangle\:AA'B, \:we\:have,}}}$

$\sf\large{\purple{\implies sin\theta = \dfrac{5}{2x} \: \: ....(ii)}}$

$\sf\large{ \: \: }$

$\sf\large{\underline{\underline{From\:(i) \:and\:(ii)}}}$

$\sf\large{\orange{\implies \dfrac{y}{x+10} = \dfrac{5}{2x} \: \: ...(iii)}}$

$\sf\large{ \: \: }$

$\sf\large{\underline{\underline{Similarly:}}}$

$\sf\large{\green{\implies \dfrac{y}{x-10} = \dfrac{15}{2x} \: \: ...(iv)}}$

$\sf\large{ \: \: }$

$\sf\large{\underline{\underline{From\:(iii)\:and\:(iv)}}}$

$\sf\large{\red{\implies 5x + 50 = 15x - 150}}$

$\sf\large{\purple{\implies 5x - 15x = -150 - 50}}$

$\sf\large{\orange{\implies -10x = -200}}$

$\sf\large{\green{\implies x = 20}}$

$\sf\large{ \: \: }$

$\sf\large{\underline{\underline{Therefore:}}}$

$\sf\large{ \: \: \: \: \: Radius\:of\: α1 = 20}$