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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:
21x2 – 4y2 = 84
Solution
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.
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