Question 616289
The form for horizontal hyperbolas is:
{{{(x-h)^2/a^2 - (y-k)^2/b^2  = 1}}}
The form for vertical hyperbolas is:
{{{(y-k)^2/a^2 - (x-h)^2/b^2 = 1}}}
In these forms, the center of the hyperbola is (h, k). and the distance from the center to a vertex is "a" and the distance from the center to a co-vertex is "b". So to write the equation we have to find:<ul><li>the center</li><li>a</li><li>b</li><li>whether the hyperbola is vertical or horizontal</li></ul>From the information we're given, perhaps the easiest one of these to find is whether the hyperbola is vertical or horizontal. The vertices have the same x coordinates, 0, but different y coordinates. This makes them up and down from each other. So this hyperbola will be vertical.<br>
Probably the next easiest is finding the center. The center is always halfway between the vertices (and halfway between the co-vertices for that matter). With vertices of (0, -5) and (0, 5) you might be able to just "see" that (0, 0) is halfway between them. This makes both "h" and "k" equal to zero. (If you can't just "see" what the center is, then just find the average of the x coordinates and find the average of the y coordinates of the vertices (or co-vertices). These averages will be the x and y coordinates of the center.)<br>
Now that we have the center we can find the "a" and the "b". The distance from the center to wither vertex is 5. (If you can't "see" what this distance is, use the distance formula to find the distance from the center to either of the vertices.) This makes a = 5.<br>
To find "b" we find how far it is from the center to a co-vertex. (Again, use the distance formula if you can't just see what this distance is.) The distance from the center to a co-vertex is 2. So b = 2.<br>
We are now ready to write the equation. Since we found that this hyperbola is vertical we will use:
{{{(y-k)^2/a^2 - (x-h)^2/b^2 = 1}}}
Substituting our "h", "k", "a" and "b" we get:
{{{(y-(0))^2/(5)^2 - (x-(0))^2/(2)^2 = 1}}}
which simplifies to:
{{{y^2/25 - x^2/4 = 1}}}