Question

PLEASE HELP!!!

Find an equation in standard form for the hyperbola with vertices at (0, ±4) and foci at (0, ±5).

Answer 1

If you plot these points on a coordinate plane, you see that both vertices and foci lie on the y axis. This means that you have a vertical hyperbola, and the equation looks like this:
`((y-k) ^(2) )/( a^(2) ) - ((x-h) ^(2) )/( b^(2) ) =1` where h and k are the center. When you look at your graph, the origin is dead center between the vertices. (0, 0) is our h and k. Now we need a, b, and c. a is the distance between the center and the vertices, so our a = 4, and c is the distance between the center and the foci, so our c = 5. Use these in Pythagorean's Theorem to solve for b:
`(4) ^(2)+ b^(2)=(5) ^(2)` and
`16+ b^(2) =25` and b = 3. So we have all we need to do is replace all the variables. Our equation then would be this one:
`((y-0) ^(2) )/(16) - ((x-0) ^(2) )/(9) =1` or, simplified,
`( y^(2) )/(16) - ( x^(2) )/(9) =1`