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Question Number 43131 by ajfour last updated on 07/Sep/18

Commented by ajfour last updated on 07/Sep/18

$Q . 43128 (A p o s s i b l e s o l u t i o n)$
	Answered by ajfour last updated on 07/Sep/18

	$A r e a = \frac{y}{2} [\sqrt{64 - {(\frac{15}{2 y} + \frac{y}{2})}^{2}} + \sqrt{121 - {(\frac{85}{2 y} + \frac{y}{2})}^{2}}]$ $w h e r e y \in [5, 15] .$
	Answered by ajfour last updated on 07/Sep/18

	$l e t p r e s e n t l y O P = y = b$ $p o i n t A i s i n t e r s e c t i o n o f$ $x^{2} + y^{2} = 64 a n d$ $x^{2} + {(y - b)}^{2} = 49$ $\Rightarrow b (2 y_{A} - b) = 15$ $y_{A} = \frac{15}{2 b} + \frac{b}{2}$ $\Rightarrow x_{A} = \sqrt{64 - {(\frac{15}{2 b} + \frac{b}{2})}^{2}} . . (i)$ $p o i n t B i s t h e i n t e r s e c t i o n o f$ $x^{2} + y^{2} = 121 a n d$ $x^{2} + {(y - b)}^{2} = 36$ $\Rightarrow b (2 y_{B} - b) = 85$ $y_{B} = \frac{85}{2 b} + \frac{b}{2}$ $\Rightarrow x_{B} = \sqrt{121 - {(\frac{85}{2 b} + \frac{b}{2})}^{2}}$ $A r e a = \frac{b}{2} (x_{A} + x_{B})$ $A r e a = \frac{b}{2} [\sqrt{64 - {(\frac{15}{2 b} + \frac{b}{2})}^{2}}$ $+ \sqrt{121 - {(\frac{85}{2 b} + \frac{b}{2})}^{2}}]$ $R e p l a c i n g b b y y$ $A r e a = \frac{y}{2} [\sqrt{64 - {(\frac{15}{2 y} + \frac{y}{2})}^{2}}$ $+ \sqrt{121 - {(\frac{85}{2 y} + \frac{y}{2})}^{2}}] .$
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