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:

21x² – 4y² = 84

Answer 1

Given equation of the hyperbola is 21x² – 4y² = 84

∴ `x^2/4 - y^2/21` = 1

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

a² = 4 and b² = 21

∴ a = 2 and b = `sqrt(21)`

Length of transverse axis = 2a = 2(2) = 4

Length of conjugate axis = 2b = `2sqrt(21)`

We know that

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

= `sqrt(4 + 21)/2`

= `sqrt(25)/2`

= `5/2`

Co-ordinates of foci are S(ae, 0) and S'(– ae, 0),

i.e., `"S"(2(5/2),0)` and `"S""'"(-2(5/2), 0)`,

i.e., S(5, 0) and S'(– 5, 0)

Equations of the directrices are x = `±"a"/"e"`.

∴ x = `±2/((5/2)`

∴ x = `± 4/5`

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

= `(2(21))/2`

= 21.