Question

A set of parallel chords of the parabola $y^2=4ax$ have their midpoints on

• A. any straight line through the vertex
• B. any straight line through the focus
• C. a straight line parallel to the axis
• D. another parabola

Answer 1

Answer: (C)

Let points $P\left(a{t}_1^2,2a{t}_1\right)$ and $Q\left(a{t}_2^2,2a{t}_2\right)$ lie on the parabola $y^2=4ax$.
Here, points $P$ and $Q$ are variable. But the slope of chord $PQ$,
$m_{PQ}=\frac{2}{t_1+t_2}$ is constant.
Now, let the midpoint of $PQ$ be $R(h,k)$. Then,
$k=\frac{2a{t}_1+2a{t}_2}{2}$
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.