Find the equation for the ellipse that satisfies the given conditions: Vertices $\left(0,\pm 13\right)$, foci $\left(0,\pm 5\right)$ 

We have vertices $$\displaystyle \left ( 0,\pm 13 \right ) $$, foci $$\left ( 0, \pm 5 \right )$$ 
Clearly here the vertices are on the $$y$$-axis 
Therefore the equation of the ellipse will be of the form $$\displaystyle

\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}= 1$$
where $$a$$ is the semi-major axis and $$\Rightarrow  a = 13, ae = 5\Rightarrow e =\cfrac{5}{13}$$
Now, we know that $$\displaystyle b^{2}=a^2(1-e^2)=a^2-a^2e^2 $$
$$\displaystyle \Rightarrow  b^{2}=13^2-5^2=144\Rightarrow b = 12$$
Thus the equation of the ellipse is $$\displaystyle

\frac{x^{2}}{12^{2}}+\frac{y^{2}}{13^{2}}= 1$$ or $$\dfrac{x^{2}}{144}+\dfrac{y^{2}}{169} = 1$$