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:

x = 2 sec θ, y = `2sqrt(3) tan theta`

Answer 1

Given equation of the hyperbola is x = 2 sec θ, y = `2sqrt(3) tan theta`

Since sec²θ – tan²θ = 1,

`(x/2)^2 - (y/(2sqrt(3)))^2` = 1

= `x^2/4 - y^2/12` = 1

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

a² = 4 and b² = 12

∴ a = 2 and b = `2sqrt(3)`

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

Length of conjugate axis = 2b = `2(2sqrt(3))=4sqrt3`

We know that

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

= `sqrt(4 + 12)/2`

= `sqrt(16)/2`

= `4/2`

= 2

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

i.e., S(2 (2), 0) and S'(– 2 (2), 0),

i.e., S(4, 0) and S'(– 4, 0)

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

∴ x = `± 2/2`

∴ x = ±1

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

= `(2(12))/2`

= 12.