Let $S$ and $S^{\prime}$ be the foci of an ellipse and $B$ be any one of the extremities of its minor axis. If $\Delta \mathrm{S}^{\prime}\mathrm{BS}$ is a right angled triangle with right angle at B and area ($\Delta \mathrm{S}^{\prime}\mathrm{BS}$) $=8$ sq. units, then the length of a latus rectum of the ellipse is :

JEE Main · 2019 · 12 Jan 2019 · Shift 2 · Ellipse

Let

S

and

S^{'}

be the foci of an ellipse and

B

be any one of the extremities of its minor axis. If

Δ S^{'} BS

is a right angled triangle with right angle at B and area (

Δ S^{'} BS

)

= 8

sq. units, then the length of a latus rectum of the ellipse is :

Answer: (A)

Let the foci be

S (\pm a e, 0)

and

B (0, b)

an end of the minor axis. A right angle at

B

means the slopes to the two foci multiply to

- 1

, giving

b^{2} = a^{2} e^{2}

. The triangle's area is

\frac{1}{2} (2 a e) (b) = a e b = 8

. Combining

b^{2} = a^{2} e^{2}

with

b^{2} = a^{2} (1 - e^{2})

gives

e^{2} = 1 - e^{2}

, so

e = \frac{1}{2}

; then

a e = b

turns

a e b = 8

into

b^{2} = 8

, and

a^{2} = \frac{b ^{2}}{e ^{2}} = 16

. The latus rectum is

\frac{2 b ^{2}}{a} = \frac{16}{4} = 4

JEE Main · 2019 · 12 Jan 2019 · Shift 2 · Single correct option