First slide

Parabola

Question

The locus of the mid-points of the chords the parabola $x^{2} = 4 p y$ having slope $m$ is a:

Moderate

A

line parallel to x-axis at a distance |2pm| from it.
B

line parallel to y-axis at a distance |2pm| from it
C

line parallel to $y = m x, m \neq 0$ at a distance |2pm| from it.
D

circle with centre at the origin and radius |2pm|.

Solution

Let $(h, k)$ be the mid point of a chord then its equation is
$h x - 2 p (y - k) = h^{2} - 4 p k$ $($ using $T = S)$
slope of this chord is $\frac{h}{2 p} = m \Rightarrow h = 2 p m$
Locus of $(h, k)$ is $x = 2 p m$
which is line parallel to y-axis at a distance |2pm| from it.

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