Question 859382
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 {{{a=7}}} ;
the focal distance would be {{{c=5}}} ,
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
{{{x^2/b^2+y^2/a^2=1}}} .
We would need to find {{{b^2}}}
We could calculate the semi-minor axis, {{{b}}} ,
from the relation {{{a^2=b^2+c^2}}} .
In the case of the ellipse centered at (0,0), with a vertex at (0,7) and a focus at (0,5),
{{{7^2=b^2+5^2}}}
{{{49=b^2+25}}}
{{{49-25=b^2}}}
{{{b^2=24}}} .
The equation of such an ellipse would be
{{{x^2/24+y^2/49=1}}} .
 
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 {{{k<5}}} ,
the lengths of the semi-major axis and the focal distance would be
{{{a=7-k}}} and {{{c=5-k}}} .
Then we would have
{{{b^2=(7-k)^2-(5-k)^2}}}
{{{b^2=49-14k+k^2-(25-10k+k^2)}}}
{{{b^2=49-14k+k^2-25+10k-k^2)}}}
{{{b^2=24-4k)}}}
and the equation would be
{{{x^2/(24-4k)+(y-k)^2/(7-k)^2=1}}}
For example, with the center at (0,2),
the equation would be
{{{x^2/16+(y-2)^2/25=1}}}