Question

The normals at three points P, Q and R of the parabola $y^2=4ax$ meet at (h, k). The centroid of the $\Delta PQR$ lies on:

• A. x = 0
• B. y = 0
• C. x = - a
• D. y = a

Answer 1

Answer: (B)

y = 0

The sum of ordinates of feet of normals drawn from a point to the parabola, $y^2=4ax$ is always zero.
Now, as normals at three points P, Q and R of parabola $y^2=4ax$ meet at (h, k).
$\Rightarrow$ The normals from (h, k) to $y^2=4ax$ meet the parabola at P, Q and R.
$\Rightarrow$ y-coordinate $y_1,y_2,y_3$ of these points and R will be zero.
$\Rightarrow$ y -coordinate of the centroid of $\Delta$PQR.
i.e., $\frac{y_1+y_2+y_3}{3}=\frac{0}{3}=0$
$\therefore$ centroid lies on $y=0$.