Ellipses: Finding the Equation from Information

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 a² 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 b² = a² − c² gives me 16 − 9 = 7 = b².

(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:

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 a² will go with the x part of the ellipse equation.

The equation b² = a² − c² gives me 9 − 4 = 5 = b², and this is all I need to create my equation:

Since the vertex is 5 units below the center, then this vertex is taller than it is wide, and the a² will go with the y part of the equation.

Also, a = 5, so a² = 25.

I know that b² = a² − c², 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.

Then my equation is:

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 a² 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 a² = 64/9.

The equation b² = a² − c² gives me:

Now that I have values for a² and b², I can create my equation: