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Question Number 48324 by mr W last updated on 22/Nov/18

Commented by ajfour last updated on 22/Nov/18

Commented by ajfour last updated on 22/Nov/18

$l e t O B = 2 a$ $e q . o f O Q : y = m x$ $e q . o f B P : y = - s (x - 2 a)$ $y_{P} = \frac{7}{a}, y_{Q} = \frac{6}{a}$ $M (\frac{2 a s}{m + s}, \frac{2 a s m}{m + s})$ $\Rightarrow (\frac{2 a s m}{m + s}) a = 4$ $\Rightarrow \frac{a^{2}}{(\frac{1}{s} + \frac{1}{m})} = 2$ $\Rightarrow \frac{1}{s} + \frac{1}{m} = \frac{a^{2}}{2} . . . . . (i)$ $x_{P} = 2 a - \frac{7}{a s}; x_{Q} = \frac{6}{a m}$ $e q . o f O A : y = \frac{(7 / a) x}{2 a - \frac{7}{a s}}$ $e q . o f A B : y = - \frac{(6 / a) (x - 2 a)}{2 a - \frac{6}{a m}}$ $y_{A} = - \frac{(6 / a) [y_{A} (2 a - \frac{7}{a s}) (\frac{a}{7}) - 2 a]}{2 a - \frac{6}{a m}}$ $\Rightarrow y_{A} [(2 a - \frac{6}{a m}) + \frac{6}{7} (2 a - \frac{7}{a s})]$ $= 12$ $\Rightarrow y_{A} = \frac{12}{(\frac{26 a}{7} - \frac{6}{a m} - \frac{6}{a s})}$ $u s i n g (i)$ $y_{A} = \frac{12}{\frac{26 a}{7} - \frac{6}{a} (\frac{a^{2}}{2})} = \frac{84}{5 a}$ $l e t A r e a (△ A O B) = △$ $a n d r e q u i r e d a r e a = A, t h e n$ $△ = A + 9$ ${a y}_{A} = A + 9$ $\Rightarrow A = \frac{84}{5} - 9 = \frac{39}{5} .$
Commented by mr W last updated on 22/Nov/18

$t h a n k s f o r t h i s a n a l y t i c a l s o l u t i o n!$
Commented by mr W last updated on 22/Nov/18

$a l t e r n a t i v e w a y :$ $l e t a r e a o f A P M Q A = x$ $\frac{A Q}{Q B} = \frac{3 + x}{4 + 2} = \frac{3 + x}{6}$ $\frac{A P}{P O} = \frac{2 + x}{4 + 3} = \frac{2 + x}{7}$ $\frac{Δ_{A M Q}}{2} = \frac{A Q}{Q B} = \frac{3 + x}{6}$ $\Rightarrow Δ_{A M Q} = \frac{3 + x}{3}$ $\frac{Δ_{A M P}}{3} = \frac{A P}{P O} = \frac{2 + x}{7}$ $\Rightarrow Δ_{A M P} = \frac{3 (2 + x)}{7}$ $Δ_{A M P} + Δ_{A M Q} = x$ $\Rightarrow \frac{3 (2 + x)}{7} + \frac{3 + x}{3} = x$ $\Rightarrow 9 (2 + x) + 7 (3 + x) = 21 x$ $\Rightarrow 5 x = 39$ $\Rightarrow x = 39 / 5$
Commented by ajfour last updated on 22/Nov/18

$I d o n t t h i n k, i c a n t h i n k l i k e t h i s!$
Commented by mr W last updated on 22/Nov/18

${y o u}^{'} r e j u s t t o o m o d e s t s i r!$
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