Question

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 1

The correct answer is 75
Equation of ellipse is: x 2 + 4 y 2 + 2 x + 8 y – λ = 0
(x + 1)2 + 4 (y + 1)2 = λ + 5
$\frac{(x+1)^2}{λ+5} + \frac{(y+1)^2}{(\frac{λ+5}{4})} = 1$
Length of latus rectum
$= \frac{2.(\frac{λ+5}{4})}{\sqrt{λ+5}} = 4$
∴ λ = 59
Length of major axis
$= 2.\sqrtλ+5 = 16 = l$
$∴ λ+l = 75$