On the previous page, we saw how to start with an equation and extract from it information about the ellipse. You'll also need to work the other way, finding the equation for an ellipse from a list of its properties. We'll cover that topic here.

Again, you would be well-advised to practice, practice, practice this material.

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- Write an equation for the ellipse having one focus at (0, 3), a vertex at (0, 4), and its center at (0, 0).

Since the focus and vertex are above and below each other, rather than side by side, I know that this ellipse must be taller than it is wide. Then *a*^{2} will go with the *y* part of the equation.

Also, since the focus is 3 units above the center, then *c* = 3; since the vertex is 4 units above, then *a* = 4.

The equation *b*^{2} = *a*^{2} − *c*^{2} gives me 16 − 9 = 7 = *b*^{2}.

(Since I wasn't asked for the length of the minor axis or the location of the co-vertices, I don't need the value of *b* itself. So I won't bother taking the square root here.)

Then my equation is:

- Write an equation for the ellipse with vertices (4, 0) and (−2, 0) and foci (3, 0) and (−1, 0).

The center is midway between the two foci, so (*h*, *k*) = (1, 0), by the Midpoint Formula.

Each focus is 2 units from the center, so *c* = 2.

The vertices are 3 units from the center, so *a* = 3.

Also, the foci and vertices are to the left and right of each other, so this ellipse is wider than it is tall, and *a*^{2} will go with the *x* part of the ellipse equation.

The equation *b*^{2} = *a*^{2} − *c*^{2} gives me 9 − 4 = 5 = *b*^{2}, and this is all I need to create my equation:

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- Write an equation for the ellipse centered at the origin, having a vertex at (0, −5) and containing the point (−2, 4).

Since the vertex is 5 units below the center, then this vertex is taller than it is wide, and the *a*^{2} will go with the *y* part of the equation.

Also, *a* = 5, so *a*^{2} = 25.

I know that *b*^{2} = *a*^{2} − *c*^{2}, but I don't know the values of *b* or *c*. However, I do have the values of *h*, *k*, and *a*, and also one value for each of *x* and *y*, those values being the coordinates of the point they gave me on the ellipse.

So I'll set up the equation with everything I've got so far, and solve for *b*.

16*b*^{2} + 100 = 25*b*^{2}

100 = 9*b*^{2}

100/9 = *b*^{2}

Then my equation is:

How did I go from the next-to-last line to the last line above? By remembering that, when we divide by a fraction, we can convert it using flip-n-multiply. This is how the 9 moved from the denominator of the fraction inside the one denominator, to being in the numerator in my answer.

- Write an equation for the ellipse having foci at (−2, 0) and (2, 0) and eccentricity
*e*= 3/4.

The center is between the two foci, so (*h*, *k*) = (0, 0).

Since the foci are 2 units to either side of the center, then *c* = 2, this ellipse is wider than it is tall, and *a*^{2} will go with the *x* part of the equation.

I know that *e* = *c*/*a*, so 3/4 = 2/*a*. Solving the proportion, I get *a* = 8/3, so *a*^{2} = 64/9.

The equation *b*^{2} = *a*^{2} − *c*^{2} gives me:

64/9 − 4 = 64/9 − 36/9 = 28/9 = *b*^{2}

Now that I have values for *a*^{2} and *b*^{2}, I can create my equation:

They might give you the picture of an ellipse, and ask you to find its equation. If they do, then they'll have to give you nice easy points on the graph (that is, points that really are where they *appear* to be). Feel safe in assuming that points that appear to have whole-number coordinates actually do.

URL: https://www.purplemath.com/modules/ellipse3.htm

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