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Question Number 206919 by BaliramKumar last updated on 30/Apr/24

	Answered by A5T last updated on 30/Apr/24

	$M A = 2 x; M C = 3 x \Rightarrow A C = 5 x$ $\Rightarrow A B + B C = 4 + 3 x + 2 x = 5 x + 4$ ${(2 + 3 x)}^{2} + {(2 + 2 x)}^{2} = 25 x^{2}$ $\Rightarrow 3 x^{2} - 5 x - 2 = 0 \Rightarrow x = 2 o r \frac{- 1}{3}$ $\Rightarrow A B + B C = 14$
	Answered by A5T last updated on 30/Apr/24

	$L e t t h e p e r p e n d e n d i c u l a r f r o m O_{1} t o A C m e e t$ $i t a t D .$ $1 - \frac{A D}{A M} = \frac{A M - A D}{A M} = \frac{D M}{4} = 1 - \frac{r_{O_{1}}}{r_{O}}$ $\Rightarrow D M = 4 - 2 r_{O_{1}}$ ${(r_{O_{1}} + r_{O})}^{2} = {(r_{O} - r_{O_{1}})}^{2} + {(D M)}^{2}$ $\Rightarrow {(r_{O_{1}} + 2)}^{2} = {(2 - r_{O_{1}})}^{2} + {(4 - 2 r_{O_{1}})}^{2}$ $\Rightarrow r_{O_{1}} = 3 - \sqrt{5}$ $S i m i l a r l y, w e g e t f o r O_{2}$ ${(r_{O_{2}} + 2)}^{2} = {(2 - r_{O_{2}})}^{2} + (6 - 3 r_{O_{2}})$ $\Rightarrow r_{O_{2}} = \frac{22 - 4 \sqrt{10}}{9}$
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