Question

If $x+y=k$ is normal to $y^2=12x$, then $k=$?

• A. $3$
• B. $6$
• C. $9$
• D. None of these

Answer 1

The correct option is C

$9$

Explanation for the correct option:

Step $1$: Find the value of $a$ and the slope $\left(m\right)$.

Given : $x+y=k$ is normal to $y^2=12x$.

We know that, $y^2=4ax$ is the equation for the standard parabola and $y=mx–2am–a{m}^3$ is the normal for the standard parabola

Comparing it with the standard parabola, we get $a=3$

Now, from $x+y=k$,

We know that slope $\left(m\right)$ is given by $\frac{dy}{dx}$

Differentiate $x+y=k$ with respect to $x$,

$$\begin{gathered} 1+\frac{dy}{dx}=0 \\ \Rightarrow \frac{dy}{dx}=-1 \\ \Rightarrow m=-1 \end{gathered}$$

Step $2$: Find the value of $k$.

Put the value of $a$ and $m$ in the equation of normal $y=mx–2am–a{m}^3$,

$\Rightarrow$ $y=-x-2\left(3\right)(-1)-3{(-1)}^3$

$\Rightarrow$ $y=-x+6+3$

$\Rightarrow$ $x+y=9$

Since $x+y=k$, thus the value of $k$ is $9$.

Hence, Option $\left(\mathrm{C}\right)$ is correct.