# 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 - Mathematics and Statistics

Sum

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

#### Solution

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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