Question 859382: How can I write the equation of an ellipse with the vertex (0,7) and focus (0,5)
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! A parabola has one focus and one vertex.
Knowing the location of focus and vertex, we can write the equation of a parabola.
An ellipse has two foci and two vertices.
As the problem is stated,
with a vertex and a focus on the y-axis,
we know that the major axis lies on the y-axis,
and that means that the center lies on the y-axis,
but we do not know the exact location of the center,
so there is not enough information to write the equation.
There are many possible ellipses with a vertex at (0,7) and a focus at (0,5).
If the ellipse were centered at (0,0), the origin,
a vertex at (0,7) and a focus at (0,5) would give you enough information.
In that case, the semi-major axis distance would be  ;
the focal distance would be  ,
and we would know that the major axis lies on the y-axis.
An ellipse centered at the origin with a major axis along the y-axis as the equation
 .
We would need to find 
We could calculate the semi-minor axis,  ,
from the relation  .
In the case of the ellipse centered at (0,0), with a vertex at (0,7) and a focus at (0,5),
 .
The equation of such an ellipse would be
 .
If the center is not specified, we cannot write the equation.
With a vertex at (0,7) and a focus at (0,5),
we know that the major axis and the center are along the y-axis,
but we do not know the y-coordinate of the center,
With a center at (0,k), with  ,
the lengths of the semi-major axis and the focal distance would be
 and  .
Then we would have
and the equation would be
For example, with the center at (0,2),
the equation would be
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