Question

Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the length of latus rectum of the hyperbola:

3x² – y² = 4

Answer 1

The equation of the hyperbola is 3x² – y² = 4

i.e. `x^2/((4/3)) - y^2/4` = 1

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

a² = `4/3`, b² = 4

∴ a = `2/sqrt(3)`, b = 2

(1) Length of transverse axis = 2a = `2(2/sqrt(3)) = 4/sqrt(3)`

(2) Length of conjugate axis = 2b = 2(2) = 4

(3) Eccentricity = e = `sqrt("a"^2 + "b"^2)/"a"`

= `sqrt(4/3 + 4)/((2/sqrt(3))`

= 2

(4) ae = `(2/sqrt(3))(2) = 4/sqrt(3)`

Coordinates of foci = (± ae, 0) = `(± 4/sqrt(3), 0)`

(5) `"a"/"e" = ((2/sqrt(3)))/2 = 1/sqrt(3)`

The equations of directrices are

x = `± "a"/"e"` i.e., x = `± 1/sqrt(3)`

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

= `(2(4))/((2/sqrt(3))`

= `4sqrt(3)`