If the equation $a x^{2} - 6 x y + y^{2} + 2 g x + 2 f y + c = 0$ represents a pair of line whose slopes are $m$ and $m^{2}$ , then sum of all possible values of $a$ is-

17

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- 19

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19

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- 17

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Solution

The correct option is A $17$
$a x^{2} - 6 x y + y^{2} + 2 g x + 2 f y + c = 0$
$= (y + 3 x + c_{1}) (y + 3^{2} x + c_{2})$
$y^{2} + m^{2} x y + c_{2} y + m x y + m^{3} x^{2} + m x c_{2} + c_{1} y + c_{1} m^{2} x + c_{2} c_{2}$
$a = m^{3}$
$m^{2} + m = - 6 \Rightarrow m^{2} + m - 6 = 0$
$a = 8, - 27$ $m^{2} + 3 m - 2 m - 6 = 0$
$m (m + 3) - 2 (m + 3) = 0$
$(m - 2) (m + 3) = 0$
$m = 2, - 3$
sum of total possible of a
= -19

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