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प्रश्न
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:
x2 – y2 = 16
उत्तर
Given equation of the hyperbola is x2 – y2 = 16
∴ `x^2/16 - y^2/16` = 1
Comparing this equation with `x^2/"a"^2 - y^2/"b"^2` = 1, we get
a2 = 16 and b2 = 16
∴ a = 4 and b = 4
Length of transverse axis = 2a = 2(4) = 8
Length of conjugate axis = 2b = 2(4) = 8
We know that
e =`sqrt("a"^2 + "b"^2)/"a"`
= `sqrt(16 + 16)/4`
= `sqrt(32)/4`
= `(4sqrt(2))/4`
= `sqrt(2)`
Co-ordinates of foci are S(ae, 0) and S'(– ae, 0),
i.e., `"S"(4sqrt(2), 0)` and `"S""'"(-4 sqrt(2), 0)`
Equations of the directrices are x = `± "a"/"e"`.
∴ x = `± 4/sqrt(2)`
∴ x = `±2sqrt(2)`
Length of latus rectum = `(2"b"^2)/"a"`
= `(2(16))/4`
= 8.
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