If the length of the latus rectum of the ellipse $x^{2}+ 4 y^{2}+2 x+8 y-\lambda=0$ is 4, and $l$ is the length of its major axis, then $\lambda+l$ is equal to ______.

JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Ellipse

If the length of the latus rectum of the ellipse

x^{2} + 4 y^{2} + 2 x + 8 y - λ = 0

is 4, and

l

is the length of its major axis, then

λ + l

is equal to ______.

Answer: 75

First, complete the square to write the ellipse in standard form:

x^{2} + 2 x + 4 y^{2} + 8 y = λ

(x + 1)^{2} - 1 + 4 (y + 1)^{2} - 4 = λ

(x + 1)^{2} + 4 (y + 1)^{2} = λ + 5

\frac{( x + 1 ) ^{2}}{λ + 5} + \frac{( y + 1 ) ^{2}}{( λ + 5 ) /4} = 1

Here,

a^{2} = λ + 5

and

b^{2} = (λ + 5) /4

, so

a > b

and the major axis is horizontal.

The length of the latus rectum is

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

.

Substituting:

\frac{2 ( λ + 5 ) /4}{λ + 5} = 4

\frac{λ + 5}{2 λ + 5} = 4

\frac{λ + 5}{2} = 4

λ + 5 = 8

λ + 5 = 64

λ = 59

The length of the major axis is

l = 2 a = 264 = 16

.

Therefore,

λ + l = 59 + 16 = 75

JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Integer answer