The normal at a point on the parabola y 2 = 4 x passes through ( 5 , 0 ) . If two more normals to this…

The given equation of a parabola,

$y^2=4x$

The equation of normal at a point $P(5,0)$

$y=-tx+2t+t^3$

$0=-5t+2t+t^3$

$3t=t^3$

$t^2=3$

$t=\pm \sqrt{3},0$

Here, $a=1$

Co-normal point

$\begin{array}{l}P\left(a{t}_1^2,2a{t}_1\right)=(3,2\sqrt{3})\end{array}$

$Q\left(a{t}_2^2,2a{t}_2\right)=(3,-2\sqrt{3})$

$R\left(a{t}_3^2,2a{t}_3\right)=(0,0)$

The centroid is,

$C=\left(\frac{3+3+0}{3},\frac{2\sqrt{3}-2\sqrt{3}+0}{3}\right)$

$C=\left(\frac{6}{3},0\right)=\left(2,0\right)$