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)

We know that, the sum of ordinates of feet of normals drawn from a point to the parabola, $y^2=4ax$ is always zero.
Norv, 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 $P_{1}Q$ and $R$.
$\Rightarrow y$-coordinates $y_1,y_2,y_3$ of these points $P,Q$ and $R$ will be zero.
$\Rightarrow y$-coordinate of the centroid of $\Delta PQR$
$i.e.,\frac{y_1+y_2+y_3}{3}=\frac{0}{3}=0$
Hence, centroid lies on $y=0$.