JEE Main · 2021 · 31 Aug 2021 · Shift 2 · Ellipse

The locus of mid-points of the line segments joining

(- 3, - 5)

and the points on the ellipse

\frac{x ^{2}}{4} + \frac{y ^{2}}{9} = 1

is:

Answer: (A)

Solution

Let

(h, k)

be the midpoint of

(- 3, - 5)

and a point

(x_{0}, y_{0})

on the ellipse, so

x_{0} = 2 h + 3

and

y_{0} = 2 k + 5

. Substituting into

\frac{x _{0}^{2}}{4} + \frac{y _{0}^{2}}{9} = 1

and multiplying by

36

gives

9 (2 h + 3)^{2} + 4 (2 k + 5)^{2} = 36

, which expands to

36 h^{2} + 16 k^{2} + 108 h + 80 k + 145 = 0

. The locus is

36 x^{2} + 16 y^{2} + 108 x + 80 y + 145 = 0

JEE Main · 2021 · 31 Aug 2021 · Shift 2 · Single correct option