If the equation $a x^{2} - 6 x y + y^{2} + b x + c x + d = 0$ represents a pair of lines whose slopes are $m$ and $m^{2}$ , then value(s) of $a$ is/are

a = - 8

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a = 8

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a = 27

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a = - 27

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Solution

The correct options are
B $a = 8$
D $a = - 27$
For $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$
$m = tan θ = ∣ ∣ ∣ ∣ \frac{2 \sqrt{h^{2} - a b}}{a + b} ∣ ∣ ∣ ∣$
Therefore for $a x^{2} - 6 x y + y^{2} + b x + c y + d = 0$
$tan θ = ∣ ∣ ∣ \frac{2 \sqrt{36 - a}}{a + 1} ∣ ∣ ∣$
$\Rightarrow \frac{2 \sqrt{36 - a}}{a + 1} = m$ or $m^{2}$
$\Rightarrow a = 8, - 27$

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