Question

this ellipse is centered at the origin. find its equation. vertices: (0,-6) and (0,6) foci: (0,-2) and (0,2) $\frac{x^{2}}{?}+\frac{y^{2}}{?}=1$ hint: the y - axis is the major axis

Explanation:

Step1: Determine the value of $a$

The vertices of the ellipse are $(0, - 6)$ and $(0,6)$. For an ellipse centered at the origin with the major - axis along the y - axis, the distance from the center to a vertex is $a$. So, $a = 6$, and $a^{2}=36$.

Step2: Determine the value of $c$

The foci of the ellipse are $(0,-2)$ and $(0,2)$. For an ellipse, the distance from the center to a focus is $c$. So, $c = 2$.

Step3: Use the relationship $c^{2}=a^{2}-b^{2}$ to find $b^{2}$

We know that $c^{2}=a^{2}-b^{2}$. Substituting $a^{2}=36$ and $c = 2$ (so $c^{2}=4$) into the equation, we get $4=36 - b^{2}$. Then, $b^{2}=36 - 4=32$.

Answer:

$\frac{x^{2}}{32}+\frac{y^{2}}{36}=1$