Question

Find the equation of an ellipse with vertices (3, 0) and (-3, 0) and foci of (2, 0) and (-2, 0).

I dont know which formula to use!!! Please help me out?

Answer 1

I know the answer to this question, its given in my textbook, but Im so confused on when to use which formula:

The Major Axis Horizontal Formula?
\[\frac{ x ^{2} }{ a ^{2} }+\frac{ y ^{2} }{ b ^{2} }\]

Or the Major Axis Vertical Formula?
\[\frac{ x ^{2} }{ b ^{2} }+\frac{ y ^{2} }{ a ^{2} }\]

Answer 2

First off, let's graph it.

Answer 3

From our formula for an ellipse, \(\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}\) we find that our center lies at \((0,0)\), therefore our equation turns into: \[\frac{x^2}{a^2}+\frac{y^2}{b^2}\]

Answer 4

And if our center is greater than zero, we use the other formula?

Answer 5

Secondly, just by graphing it, and looking at where our focus and vertices lie, we know that our ellipse is stretching horizontally, therefore \(a^2\) corresponds with \(x\) and \(b^2\) corresponds with \(y\)

Answer 6

We can now use the formula \(c^2=a^2+b^2\) to find our value of b.

Answer 7

We have our value of \(a\), and that is the distance from the center to one of the vertices along the `major` axis. Therefore, \(a=3\)

Answer 8

\(c\) is our distance from the center to one of the two foci. Therefore \(c=2\)