Question

The number of points on the parabola $y^2=x$ at which the slope of the normal drawn at the point is equal to the $x$ - coordinate of that point is

• A. $\infty$
• B. 1
• C. 2
• D. 0

Answer 1

Answer: (C)

Let the point be $\left(x_1,y_1\right)$.
$\therefore y_1^2=x_1\dots \dots \dots$ (i)
$y^2=x$
$2y\frac{dy}{dx}=1$
$\Rightarrow \frac{dy}{dx}=\frac{1}{2y}$
$\therefore$ Slope of normal at $\left(x_1,y_1\right)=\frac{-1}{{\left(\frac{dy}{dx}\right)}_{\left(x_1,y_1\right)}}$
$=\frac{-1}{\left(\frac{1}{2y_1}\right)}=-2y_1$
It is given that, $-2y_1=x_1\dots$ (ii)
From Eqs. (i) and (ii), we have
$y_1^2=-2y_1$
$\Rightarrow y_1^2+2y_1=0$
$\Rightarrow y_1\left(y_1+2\right)=0$
$\Rightarrow y_1=0,-2$
$\therefore x_1=0,4$
Required points are $(0,0),(4,-2)$.