Question

Can you please give and example of a hyperbola and ellipse with the same vertices and center that I can graph?
Then explain the similarities and differences between the graphs.

Answer 1

let's put the center of each at 0,0 to make it easy

ellipse
x^2/a^2 + y^2/b^2 = 1
we could make a and b the same, but that would be a circle and too easy
so lets make a = 2 and b = 1
x^2/4+y^2/1 = 1
then major axis along x = 2a = 4
vertex at -2,0 and + 2,0
(also at 0,1 and 0,-1
but that is not where our hyperbola will be:)
Now hyperbola opening right and left
x^2/a^2 - y^2/b^2 = 1
put in the same a and b
x^2/4 - y^2/1 = 1
(2,0) and (-2,0) still vertex, but it opens out to right and left from there