Question

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?

Answer 1

`x^2/109 + y^2/100 = 1`

Explanation:

Let `a = "major axis"`, `b = "minor axis"` and `c = 1/2*"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:

`(x-h)^2/a^2 + (y-k)^2/b^2 = 1`

Where `(h,k)` is the center of the ellipse. However, in this case, the center is `(0,0)`, 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`

Final Answer