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Q.
A set of parallel chords of the parabola $y^{2}=4 a x$ have their midpoints on
Conic Sections
Solution:
Let points $P\left(a t_{1}^{2}, 2 a t_{1}\right)$ and $Q\left(a t_{2}^{2}, 2 a t_{2}\right)$ lie on the parabola $y^{2}=4 a x$.
Here, points $P$ and $Q$ are variable. But the slope of chord $P Q$,
$m_{P Q}=\frac{2}{t_{1}+t_{2}}$ is constant.
Now, let the midpoint of $P Q$ be $R(h, k)$. Then,
$k =\frac{2 a t_{1}+2 a t_{2}}{2} $
or $ k =a\left(t_{1}+t_{2}\right)=\frac{2}{m} $
$\therefore y=\frac{2}{m}$
which is a line parallel to the axis of the parabola.