Question

$PQ$ and RS are normal chord to parabola $y^2=8x$ at $P\&R$ on the curve respectively and the points $P,Q,R,S$ are concyclic. If $A$ is vertex of parabola, then

• A. centroid of $△APR$ lies on $x$-axis
• B. centroid of $△APR$ lies on $y$-axis
• C. $PR$ is parallel to tangent at the vertex
• D. $PR$ is parallel to axis of parabola

Answer 1

Answer: (A), (C)

Equation of normal chords at $P\left(2t_1{}^2,4t_1\right)\&R\left(2t_2{}^2,4t_2\right)$ are
$y+t_{1}x-4t_1-2t_1{}^3=0\&y+t_{2}x-4t_2-2t_2{}^3=0$
Equation of any two degree curve through $P,Q,R,S$ is
$\left(y+t_{1}x-4t_1-2t_1^3\right)\left(y+t_{2}x-4t_2-2t_2^3\right)+\lambda \left(y^2-8x\right)=0$
$\Theta P,Q,R,S$ are concyclic
$\therefore 1+\lambda =t_1{t}_2\&t_1+t_2=0$
$\therefore$ centroid of $△APR$ lies on $x$-axis
& slope of $PR=\frac{4t_1-4t_2}{2t_1{}^2-2t_2{}^2}=\frac{2}{t_1+t_2}$
$\therefore PR$ is parallel to $y$-axis.