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

x^{2} − y^{2} = 16

#### Solution

Given equation of the hyperbola is x^{2} – y^{2} = 16

∴ `x^2/16 - y^2/16` = 1

Comparing this equation with `x^2/"a"^2 - y^2/"b"^2` = 1, we get

a^{2 }= 16 and b^{2} = 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)`.