Question

If the normals to the parabola $y^2=4ax$ at the ends of the latus rectum meet the parabola at $Q$ and $Q$, then $Q{Q}^{\prime }$ is

• A. $10a$
• B. $4a$
• C. $20a$
• D. $12a$

Answer 1

Answer: (D)

The ends of the latus rectum are $P(a,2a)$ and $P^{\prime }(a,2a)$.
Point $P$ has parameter $t_1=1$ and point $P^{\prime }$ has parameter $t_2=1$.
Normal at point $P$ meets the curve again at point $Q$ whose parameter
$t_1^{\prime }=-t_1-\frac{2}{t_1}=3$
Normal at point $P^{\prime }$ meets the curve again at point $Q^{\prime }$ whose parameter
$t_2^{\prime }=-t_2-\frac{2}{t_2}=3$
Hence, points $Q$ and $Q^{\prime }$ have coordinates $(9a,6a)$ and $(9a$, $6a)$, respectively.
Hence, $Q{Q}^{\prime }=12a$.