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