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Q.
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
TS EAMCET 2020
Solution:
Let the point be $\left(x_{1}, y_{1}\right)$.
$\therefore y_{1}^{2}=x_{1} \ldots \ldots \ldots$ (i)
$y^{2}=x $
$2 y \frac{d y}{d x}=1 $
$\Rightarrow \frac{d y}{d x}=\frac{1}{2 y}$
$\therefore $ Slope of normal at $\left(x_{1}, y_{1}\right)=\frac{-1}{\left(\frac{d y}{d x}\right)_{\left(x_{1}, y_{1}\right)}}$
$=\frac{-1}{\left(\frac{1}{2 y_{1}}\right)}=-2 y_{1}$
It is given that, $-2 y_{1}=x_{1} \ldots$ (ii)
From Eqs. (i) and (ii), we have
$y_{1}^{2}=-2 y_{1}$
$ \Rightarrow y_{1}^{2}+2 y_{1}=0$
$\Rightarrow y_{1}\left(y_{1}+2\right)=0 $
$\Rightarrow y_{1}=0,-2 $
$\therefore x_{1}=0,4$
Required points are $(0,0),(4,-2)$.