First slide

Parabola

Question

$A set of parallel chords of the parabola y^{2} = 4 a x have their midpoints on$

Moderate

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

Solution

$Let points P (a t_{1}^{2}, 2 a t_{1}) and Q (a t_{2}^{2}, 2 a t_{2}) lie on the$ $parabola y^{2} = 4 a x$
$Here, points P and Q are variable. But the slope of chord P Q,$
$m_{P Q} = \frac{2}{t_{1} + t_{2}} is constant.$
$Now, let the midpoint of P Q be R (h, k) . Then,$
$\begin{array}{l} k = \frac{2 a t_{1} + 2 a t_{2}}{2} \\ or k = a (t_{1} + t_{2}) = \frac{2}{m} \\ ∴ y = \frac{2}{m} \end{array}$
$which is a line parallel to the axis of the parabola.$

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