Question

Find the

1. lengths of the principal axes.
2. co-ordinates of the focii
3. equations of directrics
4. length of the latus rectum
5. distance between focii
6. distance between directrices of the ellipse:

3x² + 4y² = 12

Answer 1

The equation of the ellipse is 3x² + 4y² = 12

i.e. `x^2/4 + y^2/3` = 1

Comparing with `x^2/"a"^2 + y^2/"b"^2` = 1, we get,

a² = 4, b² = 3

∴ a = 2, b = `sqrt(3)`

∴ a > b

i. Length of major axis = 2a = 2(2) = 4

Length of minor axis = 2b = `2sqrt(3)`

ii. Eccentricity = e = `sqrt("a"^2 - "b"^2)/"a"`

= `sqrt(4 - 3)/2`

= `1/2`

∴ ae = `2 xx 1/2` = 1

∴ coordinates of foci = (± ae, 0) = (± 1, 0).

iii. `"a"/"e" = 2/((1/2))` = 4

The equations of directrices are

x = `± "a"/"e"`

∴ x = ± 4

iv. Length of latus rectum = `(2"b"^2)/"a"`

= `(2 xx 3)/2`

= 3

v. Distance between foci = 2ae

= `2 xx 2 xx 1/2`

= 2

vi. Distance between directires = `(2"a")/"e"`

= `(2 xx 2)/((1/2))`

= 2 × 4

= 8