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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`
Solution
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.
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