<p>sorry if this is confusing!</p>
<p>the equation for a hyperbola is either
(x-h)^2/a^2 - (y-k)^2/b = 1<br>
(x-h)^2/b - (y-k)^2/a = 1
whichever grouping has "a^2" under it is what determines whether the axis is horizontal or vertical. in the first one, the axis is horizontal because a is under (x-h)^2 so the x makes it horizontal. in the second one "a^2" is under (y-k)^2 so the axis is vertical. </p>
<p>the vertices are both "a" units from the center. remember that in the equation it's a^2, so you have to take the square root.
the foci are "c" unites from the center, and since that's not given in the equation you have to find it with the equation: c^2 = a^2 + b^2 and use the "a^2" and "b^2" values from the equation.</p>
<p>an ellipse is pretty similar.
the equation is either
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
(x-h)^2/b^2 + (y-k)^2/a^2
it's practically the same thing as a hyperbola, except instead of subtracting you are adding.
to find the foci "c" , you use the equation c^2 = a^2 - b^2 which is again, the same as the hyperbola equation except this time subtracting.
the length of the major axis = 2a
the length of the minor axis = 2b
so again, if y has a^2 under it, the ellipse is going to be vertical
and if the x has a^2 under it, the ellipse will be horizontal</p>
<p>i hope this helps? sorry if i explained it really badly, but i tried.</p>