Mathematics - Parabola Question with Solution | TestHub

Consider the parabola $y^2=4ax$

We have to find the locus of $\mathrm{R}(\mathrm{h},\mathrm{k})$, since Q has ordinate ' O ', ordinate of P is 2 k

Also P is on the curve, then abscissa of P is $k^2/a$

Now PQ is normal to curve

Slope of tangent to curve at any point $\frac{dy}{dx}=\frac{2a}{y}$

Hence slope of normal at point P is $-\frac{k}{a}$

Also slope of normal joining P and $\mathrm{R}(\mathrm{h},\mathrm{k})$ is $\frac{\frac{2k-k}{k^2}}{\frac{k^2}{a}-h}$

Hence comparing slopes $\frac{2k-k}{\frac{k^2}{a}-h}=-\frac{k}{a}$

Or $y^2=a(x-a)$

For $y^2=16x,a=4$, hence locus us $y^2=4(x-a)$