If the slope of one of the lines given by $a x^{2} + 2 h x y + b y^{2} = 0$ is $5$ times the other, then

5 h^{2} = 9 a b

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5 h^{2} = a b

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h^{2} = a b

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9 h^{2} = 5 a b

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Solution

The correct option is B $5 h^{2} = 9 a b$
If the slope of the two lines given by $a x^{2} + 2 h x y + b y^{2} = 0$ are $m_{1}, m_{2}$ , we have $m_{1} m_{2} = \frac{a}{b}$ and $m_{1} + m_{2} = \frac{- 2 h}{b}$
Since $m_{1} = 5 m_{2},$ we get $6 m_{2} = \frac{- 2 h}{b}$ and $5 m_{2}^{2} = \frac{a}{b}$
Using the two equations, we get $\frac{4 h^{2}}{36 b^{2}} = \frac{a}{5 b}$
$\Rightarrow 5 h^{2} = 9 a b$

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If the slope of one of the lines given by $a x^{2} + 2 h x y + b y^{2} = 0$ is $5$ times the other, then