# How do you write an equation of an ellipse in standard form given focus at (0,0) and vertices at (2, pi/2) and (8, 3pi/2)?

${x}^{2} / 16 + {\left(y + 3\right)}^{2} / 25 = 1$

#### Explanation:

The center of the ellipse can be solved by obtaining the average value of the vertices at $\left(2 , \frac{\pi}{2}\right) = \left(0 , 2\right)$and $\left(8 , \frac{3 \pi}{2}\right) = \left(0 , - 8\right)$

Center $\left(h , k\right) = \left(0 , - 3\right)$

there is a focus at (0, 0), vertex at (0, 2) and center at (0, -3) so that $c = 3$ and $a = 5$ by computation.

solve $b$:

${a}^{2} = {b}^{2} + {c}^{2}$

${5}^{2} = {b}^{2} + {3}^{2}$

${b}^{2} = 25 - 9$
${b}^{2} = 16$
and $b = 4$

the equation of the ellipse with vertical major axis is:

${\left(x - h\right)}^{2} / {b}^{2} + {\left(y - k\right)}^{2} / {a}^{2} = 1$

${\left(x - 0\right)}^{2} / {4}^{2} + {\left(y - - 3\right)}^{2} / {5}^{2} = 1$

${x}^{2} / 16 + {\left(y + 3\right)}^{2} / 25 = 1$

see the graph

graph{x^2/16+(y+3)^2/25=1[-20, 20,-10,10]}

have a nice day! from the Philippines..