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Question Number 50762 by ajfour last updated on 19/Dec/18

Commented by ajfour last updated on 19/Dec/18

$F i n d r a d i i r a n d R i n t e r m s o f a$ $a n d b .$
	Answered by mr W last updated on 20/Dec/18

	$B S = \sqrt{b^{2} + r^{2}}$ $\frac{O B}{B S} = \frac{B P}{A B}$ $\frac{b}{\sqrt{b^{2} + r^{2}}} = \frac{\sqrt{b^{2} + r^{2}} + r}{a + b}$ $\Rightarrow b^{2} + r^{2} + r \sqrt{b^{2} + r^{2}} = a b + b^{2}$ $\Rightarrow r^{2} + r \sqrt{b^{2} + r^{2}} = a b$ $\Rightarrow r \sqrt{b^{2} + r^{2}} = a b - r^{2}$ $\Rightarrow r^{2} (b^{2} + r^{2}) = a^{2} b^{2} - 2 {a b r}^{2} + r^{4}$ $\Rightarrow r^{2} (b + 2 a) = a^{2} b$ $\Rightarrow r^{2} = \frac{a^{2} b}{2 a + b}$ $\Rightarrow r = a \sqrt{\frac{b}{2 a + b}}$ $C O = 2 R = B P = r + \sqrt{b^{2} + r^{2}}$ $R = \frac{1}{2} (a \sqrt{\frac{b}{2 a + b}} + \sqrt{b^{2} + \frac{a^{2} b}{2 a + b}})$ $\Rightarrow R = \frac{\sqrt{b (2 a + b)}}{2}$
	Commented by ajfour last updated on 20/Dec/18

	$T h a n k y o u S i r, a b s o l u t e l y f i n e .$ $(s o u r c e : a j f o u r)$
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