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

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$- 27$

$- 8$

answer is A, C.

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Detailed Solution

$m$ and $m^{2}$ are the roots of the equation

${(\frac{y}{x})}^{2} - 6 (\frac{y}{x}) + a = 0$

$m + m^{2} = 6, m^{3} = a$

${(m + m^{2})}^{3} = 216$

$a + a^{2} + 18 a = 216$

$a^{2} + 19 a - 216 = 0$

$(a + 27) (a - 8) = 0$

$a = - 27, 8$

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