1. ## Ellipse

Q: The ellipse $\displaystyle D$ has equation $\displaystyle \frac{x^2}{25} + \frac{y^2}{9} = 1$ and the ellipse $\displaystyle E$ has equation $\displaystyle \frac{x^2}{4} + \frac{y^2}{9} =1$. The point $\displaystyle S$ is a focus of $\displaystyle D$ and the point $\displaystyle T$ is a focus of $\displaystyle E$. Find the length of $\displaystyle ST$.
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As the major axis for ellipse $\displaystyle E$ is the y-axis, does that mean that the focus lies on that axis too? Also, can someone draw a picture showing me what it is asking for? Thanks in advance.

2. I'm assuming you know some basics about ellipses in this solution:

For ellipse D: $\displaystyle \frac{x^2}{25} + \frac{y^2}{9} = 1$
An important equation for ellipses: $\displaystyle b^2 = a^2 (1-e^2)$ where a = 5, b =3, e = eccentricity
$\displaystyle e = \frac{4}{5}$
Focus S is $\displaystyle (\pm ae,0) = (\pm 4,0)$

For ellipse E: $\displaystyle \frac{x^2}{4} + \frac{y^2}{9} =1$
Because b is larger than a, the major axis, where the focus lies, is the y axis.
$\displaystyle a^2 = b^2(1-e^2)$ where a = 2, b = 3
$\displaystyle e = \frac{\sqrt5}{3}$
Focus T is $\displaystyle (0,\pm be) = (0,\pm \sqrt5)$

Now use the distance formula to find the distance between ST. Since there are two focus for each ellipse, I imagine you need to find the distance between each of them, but I'm pretty sure they will all be the same distance.