Question

Find the equations of the normal of the parabola $y^2=4x$ which pass through the point $\left(3,0\right)$ .

Answer 1

Here, $y^2=4x$, if compared with $y^2=4ax$ gives $a=1$

For $y^2=4ax$, equation of normal is given by

$y=mx-2am-a{m}^3$

Hence, for given parabola, equation of normal is

$y=mx-2m-m^3$

It passes through $(3,0)$

$\therefore 0=3m-2m-m^3$

$\Rightarrow m^3-m=0\Rightarrow m(m^2-1)=0$

$\therefore m=0,1,-1$.

Hence equations of normals are:

$\Rightarrow y=0$,

$\Rightarrow y=x-3$,

$\Rightarrow y=-x+3$ .