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Sequence

If x3y=y2x-...

Question

If $\frac{x}{3 y} = \frac{y}{2 x - 5 y} = \frac{6 x - 15 y}{x}$ and the expression $(- 4 x^{2} + 36 y - 8)$ takes the maximum value for $x = m$ and $y = n$ then $m + n =$

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Solution

The correct option is C $2$
$\frac{x}{3 y} = \frac{y}{2 x - 5 y} = \frac{6 x - 15 y}{x}$
$\Rightarrow \frac{x}{3 y} = \frac{y}{2 x - 5 y}$
$\Rightarrow 2 x^{2} - 5 x y = 3 y^{2} \to (1)$
$\frac{x}{3 y} = \frac{6 x - 15 y}{x}$
$x^{2} = 18 x y - 45 y^{2}$
$x^{2} - 18 x y = - 45 y^{2} \to (2)$
$15 \times (1) + (2),$
$\Rightarrow 31 x^{2} - 93 x y = 0$
$\Rightarrow 31 x = 93 y \to (3)$
expression $(f) = - 4 x^{2} + 36 y - 8$
$\Rightarrow f = - 4 x^{2} + \frac{36 \times 31 x}{93} - 8$
$f^{'} = - 8 x + 12 = 0$
$∴ x = \frac{3}{2}$ & $y = \frac{1}{2}$
$∴ x + y = 2$

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