Question

If $2x+y+\lambda =0$ is normal to the parabola $y^2=8x$, then $\lambda$ is

• A. $-24$
• B. $8$
• C. $-16$
• D. $24$

Answer 1

The correct option is A $-24$

Given parabola, $y^2=8x$
$\Rightarrow 2y\frac{dy}{dx}=8$
$\Rightarrow \frac{dy}{dx}=\frac{4}{y}$
Slope of normal of parabola $=-\frac{y}{4}$ ..... (i)
Given, $2x+y+\lambda =0$ is a equation of normal of the parabola $y^2=8x$.
$\therefore$ Slope of normal $=-2$ ...... (ii)
From Eqs. (i) and (ii), we get
$-\frac{y}{4}=-2\Rightarrow y=8$
$\therefore (8{)}^2=8x\Rightarrow x=8$
Now, putting the values of $xandy$ in the equation of normal
$2\left(8\right)+8+\lambda =0$
$\Rightarrow \lambda =-24$