Question

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

9x² – y² – 36x – 6y + 18 = 0

Answer 1

9x² – 36x – y² – 6y = – 18

9(x² – 4x) – (y² + 6y) = – 18

9(x – 2)² – (y + 3)² = – 18 + 36 – 9

9(x – 2)² – (y + 3)² = 9

`(x - 2)^2/1 - (y + 3)^2/9` = 1

It is hyperbola.

The transverse axis is parallel to x axis.

a² = 1, b² = 9

a = 1, b = 3

c² = a² + b²

= 1 + 9

= 10

c = `sqrt(10)`

ae = `sqrt(10)`

e = `sqrt(10)`

Centre (h, k) = (2, – 3)

Vertices (h ± a, k) = (2 ± 1, – 3)

= (2 + 1, –3) and (2 – 1, – 3)

= (3, – 3) and (1, – 3)

Foci (h ± c, k) = `(2 +-  sqrt(10), -3)`

= `(2 + sqrt(10), - 3)` and `(2 - sqrt(10), -3)`

Directrix x = `+-  "a"/"e" + "h"`

= `+-  1/sqrt(10) + 2` and x = `- 1/sqrt(10) + 2`