Question

Identify the type of conic and find centre, foci, vertices, and directrices of the following:

`(y - 2)^3/25 + (x + 1)^2/16` = 1

Answer 1

It is a hyperbola.

The transverse axis is parallel to y axis.

a² = 25, b² = 16

a = ± 5, b = 4

c² = a² + b²

= 25 + 16

= 41

c = `sqrt(41)`

ae = `sqrt(41)`

5e = `sqrt(41)`

e = `sqrt(41)/5`

Centre (h, k) = (– 1, 2)

Vertices (h, ± a + k) = (– 1, ± 5 + 2)

= (– 1, 5 + 2) and (– 1, – 5 + 2)

= (– 1, 7) and (– 1, – 3)

Foci (h, ± c + k) = `(- 1 +-  sqrt(41) + 2)`

= `(-1, sqrt(41) + 2)` and `(-1, - sqrt(41) + 2)`

Directrix x = `+-  "a"/"e" + "k"`

= `+-  5/(sqrt(41)/5) + 2`

= `+-  25/sqrt(41) + 2`

y = `25/sqrt(41) + 2` and y = `- 25/sqrt(41) + 2`