JEE Main · 2021 · 24 Feb 2021 · Shift 1 · Parabola

The locus of the mid- point of the line segment joining the focus of the parabola

y^{2} = 4 a x

to a moving point of the parabola, is another parabola whose directrix is:

Answer: (A)

Solution

The parabola

y^{2} = 4 a x

has focus

(a, 0)

, and a general point on it may be written

(a t^{2}, 2 a t)

. The midpoint of the segment joining the focus to this point is

(\frac{a + a t ^{2}}{2}, a t)

. Writing

h = \frac{a ( 1 + t ^{2} )}{2}

and

k = a t

and eliminating

t

through

t = \frac{k}{a}

gives

k^{2} = 2 a (h - \frac{a}{2})

, a parabola

y^{2} = 2 a (x - \frac{a}{2})

with vertex

(\frac{a}{2}, 0)

and semi-latus-rectum

\frac{a}{2}

. Its directrix lies a distance

\frac{a}{2}

to the left of the vertex, at

x = \frac{a}{2} - \frac{a}{2} = 0

JEE Main · 2021 · 24 Feb 2021 · Shift 1 · Single correct option