Question

The eccentricity, foci and the length of the latus rectum of the ellipse x² + 4y² + 8y – 2x + 1 = 0 are respectively equal to ______.

पर्याय

• `sqrt(3)/2;(1 +- sqrt(3), -1);2`

• `sqrt(3)/2;(1 +- sqrt(3), 1);1`

• `sqrt(3)/2;(1 +- sqrt(3), -1);1`

• `sqrt(3)/2;(1 +- sqrt(3), 1);2`

Answer 1

The eccentricity, foci and the length of the latus rectum of the ellipse x² + 4y² + 8y – 2x + 1 = 0 are respectively equal to `underlinebb(sqrt(3)/2;(1 +- sqrt(3), -1);1)`.

Explanation:

Given equation of ellipse is x² + 4y² + 8y – 2x + 1 = 0

⇒ (x – 1)² + 4(y² + 2y) = 0

⇒ `(x - 1)^2/4 + ("y" + 1)^2/1` = 1

∴ Eccentricity of ellipse is given by

e = `sqrt(1 - "b"^2/"a"^2)`

= `sqrt(1 - 1/4)`

= `sqrt(3/4)`

= `sqrt(3)/2`

Foci of the ellipse are given by (1 ± ae, –1)

Where ae = `sqrt("a"^2 - "b"^2)`

⇒ ae = `sqrt(4 - 1) = sqrt(3)`

⇒ Foci are `(1 +- sqrt(3), -1)`

Latus rectum of the ellipse is given by

= `(2"b"^2)/"a" = (2 xx 1)/2`

= 1