Question

How do you find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola `y^2/18-x^2/20=1`?

Answer 1

The vertices are `(0,3sqrt2)` and `(0,-3sqrt2)`
The foci are `(0,sqrt38)` and `(0,-sqrt38)`
the asymptotes are `y=(3x)/sqrt10` and `(-3x)/sqrt10`

Explanation:

Looking at the equation, this is an up-down hyperbola

and, we compare to the standard equation
`(y-k)^2/b^2-(x-h)^2/a^2=1`

The center is `(h,k)=(0,0)`

The vertices are `(h,k+b)=(0,sqrt18)` and `(h.k-b)=(0,-sqrt18)`

The slopes of the asymptotes are `b/a=sqrt18/sqrt20=3/sqrt10` and `-b/a=-sqrt18/sqrt20=-3/sqrt10`
The equations of the asymptotes are `y=k+-b/a(x-h)`

The equations of the asymptotes are `y=0+3x/sqrt10` and `y=0-3x/sqrt10`

To calculate the foci, we need `c=+-sqrt(a^2+b^2)=+-sqrt(18+20)=+-sqrt38`
The foci are `h,k+-c`

And the foci are `(0,sqrt38)` and `(0,-sqrt38)`