If $x + y = k$ is a normal to the parabola $y^{2} = 12 x$ , then it touches the parabola

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$y^{2} = - 36 x$

$y^{2} = - 12 x$

$y^{2} = - 9 x$

none of these

answer is A.

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Detailed Solution

Since $y = m x - 2 a m - a m^{3}$ is a normal to the parabola

$y^{2} = 4 a x,$ taking $a = 3$ and $m = - 1$

$y = - x - 2 (3) (- 1) - 3 (- 1)^{3}$

$\Rightarrow x + y = 9$ is a normal to the parabola $y^{2} = 12 x$ .

Suppose it touches the parabola $y^{2} = 4 a x$
Equation of a tangent to the parabola $y^{2} = 4 a x$ is

$y = m x + a / m$

If it represents the line $x + y = 9$ then

$m = - 1$ and $a / m = 9 \Rightarrow a = - 9$

So an equation of the required parabola is $y^{2} = 4 (- 9) x$ $\Rightarrow$

$y^{2} = - 36 x$

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