# How do you find the standard form of the equation of the ellipse given the properties foci (+-3,0), length of the minor axis 10?

May 31, 2017

${x}^{2} / 109 + {y}^{2} / 100 = 1$

#### Explanation:

Let $a = \text{major axis}$, $b = \text{minor axis}$ and $c = \frac{1}{2} \cdot \text{focal length}$

$b = 10$ is given.

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Since our foci are both $3$ away from the origin and lie on the x-axis, the center of the ellipse will be at the origin. We also know that $c = 3$, then.

Using the property of ellipses that:

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

We can determine the value of $a$, the major axis.

${a}^{2} = {10}^{2} + {3}^{2}$

${a}^{2} = 109$

Let's stop there, since our final equation relies on ${a}^{2}$ rather than $a$.

Now, we have everything we need to make the standard form equation for the given ellipse. Here is what standard form looks like for an ellipse with a horizontal major axis:

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

Where $\left(h , k\right)$ is the center of the ellipse. However, in this case, the center is $\left(0 , 0\right)$, so we don't even need to worry about $h$ and $k$.

Therefore, this specific ellipse's equation is:

${x}^{2} / 109 + {y}^{2} / 100 = 1$