If a line $2 x + 3 y + k = 0$ touches then parabola $x^{2} = 108 y$ then $k =$

27

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36

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Solution

The correct option is D $36$
$2 x + 3 y + k = 0$

$⟹ y = \frac{- (2 x + k)}{3}$

$x^{2} = 108 y = (108) \frac{- 2 k - k}{3} = - 36 (2 x + k$ )

$x^{2} + 72 x + 36 k = 0$

Here for two curves to touch each other,

So,

$72^{2} - 4 \times 1 \times 36 k = 0$

$⟹ k = \frac{72^{2}}{4 \times 36} = 36$

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