Question

Find an equation in standard form for the hyperbola with vertices at (0, ±6) and foci at (0, ±9).

Answer 1

@phi

Answer 2

the center is the average of the vertices

Answer 3

So (0,0)

Answer 4

the vertices are on the y-axis, so the hyperbola looks like a frown/smile combo
that means the "y' goes first in the standard equation
\[ \frac{(y-k)^2}{a^2}- \frac{(x-h)^2}{b^2}=1\]
a is the distance from the center to the focus
"c" is the distance from the center to the vertex
we use a and c to find b
a^2 + b^2 = c^2

we already know (h,k) is (0,0)

Answer 5

oops, got that swapped:
a is the distance from the center to the vertex
"c" is the distance from the center to the focus

Answer 6

alright.

Answer 7

so a is 6 and c is 9

Answer 8

yes, but in the equation you use a^2 = 36 and c^2=81
what is b^2 ?

Answer 9

\[3\sqrt{13}\]

Answer 10

b^2 is 117

Answer 11

a^2 + b^2 = c^2
36 + b^2 = 81

Answer 12

oh my bad

Answer 13

b^2 = 45 b=3sqrt5

Answer 14

now fill in the numbers to get the equation

Answer 15

thanks. I still have a few more