Answer the following:
Find the
(i) lengths of the principal axes
(ii) co-ordinates of the foci
(iii) equations of directrices
(iv) length of the latus rectum
(v) Distance between foci
(vi) distance between directrices of the curve
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
i. Length of transverse axis = 2a = 2(4) = 8
Length of conjugate axis = 2b = 2(4) = 8
ii. 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)`
iii. Equations of the directrices are x = `± "a"/"e"`.
∴ x = `± 4/sqrt(2)`
∴ x = `±2sqrt(2)`
iv. Length of latus rectum = `(2"b"^2)/"a"`
= `(2(16))/4`
= 8
v. Distance between foci = 2ae = `2(4)(sqrt(2)) = 8sqrt(2)`
vi. Distance between directrices = `(2"a")/"e"`
= `(2(4))/sqrt(2)`
= `4sqrt(2)`.
