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Question Number 217495 by Hanuda354 last updated on 14/Mar/25

	Answered by mr W last updated on 17/Mar/25

	Commented by Hanuda354 last updated on 18/Mar/25

	$Thanks a lot, Prof .$
	Commented by mr W last updated on 17/Mar/25

	$e q n . o f P R :$ $y = \frac{7 x}{2 (a - 3)}$ $e q n . o f S W :$ $y = a - \frac{x}{2}$ $y_{Q} = a - \frac{x_{Q}}{2} = \frac{7 x_{Q}}{2 (a - 3)}$ $\Rightarrow x_{Q} = \frac{2 a (a - 3)}{(a + 4)}$ $A = \frac{a}{2} \times \frac{2 a (a - 3)}{(a + 4)} = \frac{a^{2} (a - 3)}{a + 4}$ $A + B = \frac{(2 a - 6) (a + a - 7)}{2} = (a - 3) (2 a - 7)$ $\frac{A + B}{A} = 1 + \frac{B}{A} = 1 + \frac{41}{50} = \frac{91}{50}$ $\frac{(a - 3) (2 a - 7)}{\frac{a^{2} (a - 3)}{a + 4}} = \frac{91}{50}$ $9 a^{2} + 50 a - 1400 = 0$ $a = \frac{- 50 + \sqrt{50^{2} + 4 \times 9 \times 1400}}{18} = 10$ $S W = \sqrt{a^{2} + {(2 a)}^{2}} = 10 \sqrt{5}$ $\frac{S Q}{S W} = \frac{x_{Q}}{2 a} = \frac{a - 3}{a + 4} = \frac{10 - 3}{10 + 4} = \frac{1}{2}$ $Q W = S Q = \frac{10 \sqrt{5}}{2} = 5 \sqrt{5}$ $\frac{P Q}{P R} = \frac{x_{Q}}{2 a - 6} = \frac{a}{a + 4} = \frac{10}{14} = \frac{5}{7}$ $\Rightarrow \frac{Q R}{P R} = \frac{2}{7}$ $\frac{a}{a} \times \frac{x}{y} \times \frac{Q R}{R P} = 1$ $\frac{x}{y} \times \frac{2}{7} = 1$ $\Rightarrow \frac{x}{y} = \frac{7}{2}$ $\Rightarrow \frac{x}{x + y} = \frac{7}{9}$ $\Rightarrow x = \frac{7 \times Q W}{9} = \frac{7 \times 5 \sqrt{5}}{9} = \frac{35 \sqrt{5}}{9} ✓$
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