Question

Let $P,Q,R$ be three points on a parabola, normals at which are concurrent. The centroid of the $\Delta PQR$ must lie on

• A. a line parallel to the directrix
• B. the axis of the parabola
• C. a line of slope $1$ passing through the vertex
• D. none of these

Answer 1

The correct option is B the axis of the parabola

Equation of normal in parametric form to $y^2=4ax$ is $y=xt+2at+2a{t}^3$
Let it passes through $(h,k)$
Then, $2a{t}^3+t(h+2a)-k=0$
Sum of the roots: $t_1+t_2+t_3=0$
Now, centroid $G=(\frac{a{t_1}^2+a{t_2}^2+a{t_3}^2}{3},\frac{2a{t}_1+2a{t}_2+2a{t}_3}{3})=(\frac{a{t_1}^2+a{t_2}^2+a{t_3}^2}{3},0)$
Therefore, $G$ lies on axis of parabola