The path described here is an ellipse represented as follows:

Comparing to a standard form or an ellipse, we get $2a = 10 \rightarrow a = 5$ and $2c = 8 \rightarrow c = 4$

We know that $c^2 = a^2 - b^2 \rightarrow b = \sqrt{25-16} = \sqrt 9 = 3$

Therefore, the equation of the path traced $ = \large\frac{x^2}{25}$$+\large\frac{y^2}{9}$$ = 1$