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#?

1 Answer
Oct 26, 2016

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)#