Answer
We consider an ellipse centered at the origin whose major axis is vertical. The equation of this ellipse in standard form indicates that ${{a}^{2}}=9$ and ${{b}^{2}}=4$. Therefore, ${{c}^{2}}=\underline{5}$. The foci are located at $5,\left( 0,-\sqrt{5} \right)$ and $\left( 0,\sqrt{5} \right)$.
Work Step by Step
It is shown that ${{a}^{2}}=9$ and ${{b}^{2}}=4$
The equation is in the standard form of an ellipse’s equation, with ${{a}^{2}}=9$ and ${{b}^{2}}=4$. Also, it is given in the problem that the major axis is vertical.
The general equation of an ellipse is:
$\frac{{{x}^{2}}}{{{b}^{2}}}+\frac{{{y}^{2}}}{{{a}^{2}}}=1$
The foci are located at the points $\left( 0,c \right)$ and $\left( 0,c \right)$
Where, ${{c}^{2}}={{a}^{2}}{{b}^{2}}$.
Substitute the value of ${{a}^{2}}$ and ${{b}^{2}}$ in ${{c}^{2}}={{a}^{2}}{{b}^{2}}$ to obtain the value of $c$ and simplify as given below:
$\begin{align}
& {{c}^{2}}={{a}^{2}}{{b}^{2}} \\
& {{c}^{2}}=9-4 \\
& {{c}^{2}}=5 \\
& c=\sqrt{5}
\end{align}$
Hence, the foci ${{F}_{1}}\text{ and }{{F}_{2}}$ are $\left( 0,-\sqrt{5} \right)$ and $\left( 0,\sqrt{5} \right)$.
