Find k, if the sum of the slopes of the lines represented by $2 x^{2} + k x y - 9 y^{2} = 0$ is five times their product.

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Solution

Comparing given equation with $a x^{2} + 2 h x y + b y^{2} = 0$
$a = 2, h = \frac{k}{2}, b = - 9$

Now, Let $m_{1}$ and $m_{2}$ be the slope of the lines represented by $2 x^{2} + 2 h x - 9 y^{2} = 0$

$m_{1} + m_{2} = \frac{- 2 h}{b} = \frac{- k}{- 9} = \frac{k}{9} . . . . . . . . . . . . . . . . . . (1) m_{1} m_{2} = \frac{a}{b} = - \frac{2}{9} . . . . . . . . . . . . . . . . . . (2)$

Now, $m_{1} + m_{2} = 5 m_{1} m_{2} \Rightarrow \frac{k}{9} = 5 * \frac{- 2}{9} \Rightarrow k = - 10$

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