Question

From point $P (6,0)$ three normals are drawn to the parabola $y ^2=8 x$ and points $A , B$ and $C$ are co-normal points. Then which of the following is/are correct?

• A. Area of $\triangle ABC$ is 8
• B. Circum-centre of $\triangle ABC$ is $(5,0)$
• C. Centroid of $\triangle ABC$ is $\left(\frac{4}{3}, 0\right)$
• D. $\triangle ABC$ is a right angled triangle

Answer 1

Answer: (A), (B), (C)

Let the normal be $y=m x-4 m-2 m^3$
$\Rightarrow 0=6 m -4 m -2 m ^3 \Rightarrow m =0,1,-1 $
$A (0,0) ; B (2,4) ; C (2,-4)$
$\text { Area }=8$
$\text { Centroid } \equiv\left(\frac{4}{3}, 0\right), \text { circumcentre } \equiv(5,0) $