SOLUTION: write the equation. 1. hyperbola with foci (0,4) and (0,-4) and asymptotes at y=2x 2. circle passing through the points (12,1) and (2,-3) with center on the line 2x-5y+10=0

Question 1035355: write the equation.
1. hyperbola with foci (0,4) and (0,-4) and asymptotes at y=2x
2. circle passing through the points (12,1) and (2,-3) with center on the line 2x-5y+10=0
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
1.The major axis is on the line connecting the foci, which is the y-axis.
The center of the hyperbola is halfway between the foci, at (0,0), the origin,
so the minor axis (perpendicular to the y-axis through the center) is the x-axis.
The equation for a parabola centered at the origin, with the y-axis as its major axis is
with and .
The focal distance, , is half the distance between the foci.
In this case, it is .
The semimajor axis, , is the distance between the center and each vertex.
The major axis is the vertical segment connecting vertices (0,-a), (0,a),
which are on the major axis, between the foci, so .
The semi-minor axis is .
The asymptotes cross at the center of the hyperbola (in this case (0,0), the origin),
and have equations and .
So far, we have

The values and determine a rectangle,
passing through the vertices,
with sides measuring and ,
and diagonals measuring .
The axes and the asymptotes divide that rectangle into 8 right trianlges with legs and , and hypotenuse , so
.
Since the asymptotes have equations and ,
in this case --> --> ,
and since ,

.
So, the equation for the hyperbola in this problem is
,
or ,
or ,
or .
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