MathematicsIf x+y=k is a normal to the parabola y2=12x, find the value of k.

If $x + y = k$ is a normal to the parabola $y^{2} = 12 x$ , find the value of $k$ .

Given,

$L : y = - x + k$

Here, $m = - 1$ and $c = k$

The line $L$ will be a normal to the parabola, if

$\begin{array}{l} c = - 2 a m - a m^{3} \\ k = - 2 \times 3 \times (- 1) - 3 \times (- 1)^{3} \\ k = 6 + 3 \\ k = 9 \end{array}$

Hence, the value of $k$ is $9$ .

JEE