JEE Main · 2021 · 27 Jul 2021 · Shift 1 · Ellipse

A ray of light through

(2, 1)

is reflected at a point

P

on the

y

-axis and then passes through the point

(5, 3)

. If this reflected ray is the directrix of any ellipse with eccentricity

\frac{1}{3}

and the distance of the nearer focus from this directrix is

\frac{8}{53}

, then the equation of the other directrix can be :

Answer: (C)

Solution

Reflection at the

y

-axis sends

(2, 1)

(- 2, 1)

, so the reflected ray is the line through

(- 2, 1)

and

(5, 3)

, of slope

\frac{3 - 1}{5 + 2} = \frac{2}{7}

2 x - 7 y + 11 = 0

For an ellipse of semi-major axis

a

and eccentricity

e = \frac{1}{3}

, the distance from a focus to the directrix on its own side is

\frac{a}{e} - a e = a (3 - \frac{1}{3}) = \frac{8 a}{3} = \frac{8}{53} \Rightarrow a = \frac{3}{53}

The two directrices are parallel and

\frac{2 a}{e} = 6 a = \frac{18}{53}

apart. Writing the other one as

2 x - 7 y + λ = 0

\frac{∣ λ - 11 ∣}{4 + 49} = \frac{18}{53} \Rightarrow ∣ λ - 11 ∣ = 18

λ = 29

λ = - 7

, and the other directrix is

2 x - 7 y + 29 = 0

2 x - 7 y - 7 = 0

JEE Main · 2021 · 27 Jul 2021 · Shift 1 · Single correct option