SOLUTION: I need to find the equation of a hyperbola with foci at (-3,7) and (-3,-5) whose transverse axis is 8 units long. I have no idea where to even start with this, but I have graphed t

Question 874663: I need to find the equation of a hyperbola with foci at (-3,7) and (-3,-5) whose transverse axis is 8 units long. I have no idea where to even start with this, but I have graphed the points given.
Found 2 solutions by KMST, josgarithmetic:
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
We realize that both foci have .
That means that the center, foci and vertices have ,
and that the transverse axis, connecting the vertices is part of the vertical line .
The distance between the foci is the difference in their y-coordinates,
.
The center is midway between the foci.
The distance between each focus and the center is the focal distance,

The center is the midpoint of the segment connecting the foci,
so its coordinates are averages of the foci's coordinates:
and
So the center is at (-3,1).
That means that and appear in the equation of the hyperbola.

Since the distance between the vertices is , the length of the transverse axis,
each vertex is units away (above and below) the center.
Their coordinates of the vertices are units away from the of the center,
,
and for the other points on the hyperbola,
the coordinates are farther away from the of the center.
So <--><--> .
The equation will be
<---> . .
The way to find is through the relationship .
Substituting the known values and
-->-->-->
Substituting that value, the equation is
.

Answer by josgarithmetic(39620) (Show Source): You can put this solution on YOUR website!
Transverse Axis is the segment connecting the vertices. Yours is a length of 8 units, so a=4. Distance from center to either vertex is 4 units, half of the transverse axis length.

Focal length: . Again using focus information, the center, y value is the average of the focus y values: .
The center is (-3,1).

The negative sign is on the term for x, since your foci are vertically arranged. If the hyperbola were standard position, vertices would be on the y axis. In your case, center is NOT at the origin.

Now you have center, a, and c. You can say,
.
You still want the b value.

A substitution is often used in deriving the equation of a hyperbola.
The effect is the relationship
.
You example has .

Finally, your standard form hyperbola is
RELATED QUESTIONS
Find the equation (hyperbola) foci at ( - 3, 1) and (7, 1) and length of the... (answered by MathLover1)
If foci of a hyperbola are (0,±5) and length of semi transverse axis is 3 units, then... (answered by ikleyn)
Find the equation of hyperbola, foci at (3,5) and (3,-5), length of transverse axis is... (answered by MathLover1)
find the equation of the hyperbola with center at (4,-1) transverse axis parallel to the... (answered by MathLover1,ikleyn)
foci at ( - 3, 1) and (7, 1) and length of the transverse axis is 4... (answered by ikleyn,Edwin McCravy)
Write an equation of a hyperbola with foci (6,0) and (-6,0) if the length of the... (answered by josgarithmetic)
I need to find the standard equation of a hyperbola with transverse axis at y=-5 with a... (answered by lwsshak3)
Find the equation for the hyperbola with foci (4,6) and (-8,6) and a transverse axis... (answered by lwsshak3)
**Sorry if this appears twice, it doesn't seem to have worked the first time. A... (answered by solver91311)