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Q.
If the normals to the parabola $y^{2}=4 a x$ at the ends of the latus rectum meet the parabola at $Q$ and $Q$, then $Q Q^{\prime}$ is
Conic Sections
Solution:
The ends of the latus rectum are $P(a, 2 a)$ and $P^{\prime}(a, 2 a)$.
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 $(9 a, 6 a)$ and $(9 a$, $6 a)$, respectively.
Hence, $Q Q^{\prime}=12 a$.