Answer
Ellipse:
- the vertices are $(1\pm \frac{5}{2}, \frac{1}{5})$, $(1, \frac{1}{5}\pm1)$
- the foci are $(1\pm \sqrt{21/4},1/5)$
- the center is $(1,1/5)$
Work Step by Step
By completing the square, we can write the equation
$$
4 x^{2}+25 y^{2}-8 x-10 y=20
$$
in the form
$$
4 (x-1 )^2+25 (y-\frac{1}{5})^2=25.
$$
Then we get
$$
\left(\frac{x-1}{ 5/2}\right)^{2}+\left(\frac{y-\frac{1}{5}}{1}\right)^{2}=1
$$
which is an ellipse with $a=5/2, b=1$ and hence $ c=\sqrt{a^2-b^2}=\sqrt{21/4} $
So, we have:
- the vertices are $(1\pm \frac{5}{2}, \frac{1}{5})$, $(1, \frac{1}{5}\pm1)$
- the foci are $(1\pm \sqrt{21/4},1/5)$
- the center is $(1,1/5)$
