Given:

\[Vertices\text{ }\left( 0,~\pm ~13 \right)\text{ }and\text{ }foci\text{ }\left( 0,~\pm \text{ }5 \right)\]

Here, the vertices are on the \[x-pivot.\]

Along these lines, the condition of the circle will be of the structure\[{{x}^{2}}/{{b}^{2}}~+\text{ }{{y}^{2}}/{{a}^{2}}~=\text{ }1,~\], where ‘a’ is the semi-significant pivot.

Then, at that point, \[a\text{ }=13\text{ }and\text{ }c\text{ }=\text{ }5.\]

It is realized that \[{{a}^{2}}~=\text{ }{{b}^{2~}}+\text{ }{{c}^{2}}.\]

\[{{13}^{2}}~=\text{ }{{b}^{2}}+{{5}^{2}}\]

\[169\text{ }=\text{ }{{b}^{2}}~+\text{ }15\]

\[{{b}^{2}}~=\text{ }169\text{ }\text{ }125\]

\[b\text{ }=~\surd 144\]

\[=\text{ }12\]

∴ The condition of the circle is \[{{x}^{2}}/{{12}^{2}}~+\text{ }{{y}^{2}}/{{13}^{2}}~=\text{ }1\text{ }or\text{ }{{x}^{2}}/144\text{ }+\text{ }{{y}^{2}}/169~=\text{ }1\]