You'll also need to work the other way,
finding the equation for an ellipse from a list of its properties.

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.) 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 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:

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 a set of values for x
and y,
those values being 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.

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:

Stapel, Elizabeth.
"Conics: Ellipses: Finding the Equation from Information."
Purplemath. Available from https://www.purplemath.com/modules/ellipse3.htm.
Accessed