In an earlier lesson, the equation of an ellipse with center
at the origin and foci on the $x$axis was derived, in great detail.
You may want to review this earlier lesson
before studying the ‘inanutshell’ derivation here.
Although it is much shorter, this derivation should look strikingly familiar to
the earlier derivation.
Notice also that the variables $\,a\,$, $\,b\,$ and $\,c\,$ have the same meaning as in the earlier derivation,
and the relationship between these three variables is the same.



Summary: The equation of an ellipse with center at the origin and foci along the $\,y$axis is $$ \cssId{s58}{\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1} $$ where:


Once we have the equation for foci on the $x$axis $\displaystyle\,\left(\frac{x^2}{a^2} +\frac{y^2}{b^2} = 1\right)\,$,
there are shorter and easier ways to get the equation for foci on the $y$axis $\displaystyle\,\left(\frac{x^2}{b^2} +\frac{y^2}{a^2} = 1\right)\,$.
We didn't really have to go through this derivation again (although it's good practice).
Here are two ways.
For ease of reference, let $\,\cal G\,$ denote the graph of $\displaystyle\,\frac{x^2}{a^2} +\frac{y^2}{b^2} = 1\,$.
The two transformations used below are discussed in greater detail in this earlier optional lesson.
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
IN PROGRESS 