The sun is one of the foci of the elliptical orbit and the graph for the elliptical orbit of the planet is given below:

From the figure, observe that the given graph is an ellipse that is centered at the origin.

So, the equation of an ellipse centered at the origin is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\dots \left(1\right)$.

From the figure, observe that the major axis of an ellipse is horizontal.

From the figure, conclude that the distance from

the center to the vertex is $25$.

So, $a=25$which implies $a^2=625$.

The minor axis is vertical and the distance from the center to the endpoints of the minor axis are unknown.

To find $b$,use the location of the Sun.

Since the Sun is a focus of the ellipse at the point $(5,0)$, $c=5$.

Apply the formula: $c^2=a^2-b^2\dots \left(2\right)$

Substitute the value of $a$and $c$into the equation $\left(2\right)$.

$\begin{array}{rcl}{5}^2& =& {25}^2-b^2\\ 25& =& 625-b^2\\ b^2& =& 625-25\\ b^2& =& 600\end{array}$

Substitute the values of $a^2$and $b^2$into the equation $\left(1\right)$.

Thus, the equation of an ellipse is as follows:

$\frac{x^2}{625}+\frac{y^2}{600}=1$