Given the equation 9y^2 - x^2 = 9 determine:

a) The Center C

b) The 2 Vertices

c) The slopes of the asymptotes (enter as a reduced fraction)

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The **standard form of the equation of a hyperbola **with center (h, k) is

`(x-h)^2/a^2-(y-k)^2/b^2=1` (transverse axis is horizontal)

`(y-k)^2/a^2-(x-h)^2/b^2=1` (transverse axis is vertical)

The given equation is:

`9y^2 - x^2 = 9`

Dividing both sides by 9,

`y^2/1 - x^2/9 = 1`

`rArr (y-0)^2/1^2-(x-0)^2/3^2= 1`

This is the equation of a hyperbola whose transverse axis is vertical.

a) Its centre **C is at (0,0)**.

b) The vertices are at fixed distance a from the center (vertical transverse axis).

Here, a=1 unit. Hence the coordinates of the vertices are (0,+-a) i.e. **(0, 1) and (0, -1)**.

c) Equations of the asymptotes

Using a=1, b=3, the equations of the asymptotes are:

`y=k+-a/b(x-h)`

i.e. `y=1/3x`

and, `y=-1/3x`

**Slopes of these asymptotes are thus, 1/3 and -1/3.**

**Sources:**

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