Write the equation of the ellipse with center (5, -4), a vertex at (-5, -4) and b = 8

((x - 5)^2)/(100) + ((y + 4)^2)/(64) = 1

Write the equation of the ellipse with center (-2, 8), a co-vertex at (-2, 14) and a = 11

((x + 2)^2)/(121) + ((y - 8)^2)/(36) = 1

Write the equation of the ellipse with center (-6, -1), a vertex at (2, -1) and b = 5

((x + 6)^2)/(64) + ((y + 1)^2)/(25) = 1

Write the equation of the ellipse with center (11, 1), a co-vertex at (4, 1) and a = 12

((x - 11)^2)/(49) + ((y - 1)^2)/(144) = 1

Write the equation of the ellipse with center (1, -4), a vertex at (1, 11) and b = 14

((x - 1)^2)/(196) + ((y + 4)^2)/(225) = 1

Write the equation of the ellipse with center (10, -2), a co-vertex at (10, -9) and a = 12

((x - 10)^2)/(144) + ((y + 2)^2)/(49) = 1

Write the equation of the ellipse with center (-5, 8), a vertex at (-5, 21) and b = 6

((x + 5)^2)/(36) + ((y - 8)^2)/(169) = 1

Write the equation of the ellipse with center (5, -11), a vertex at (5, -23) and b = 8

((x - 5)^2)/(64) + ((y + 11)^2)/(144) = 1

Write the equation of the ellipse with center (5, 8), a vertex at (5, -2) and b = 9

((x - 5)^2)/(81) + ((y - 8)^2)/(100) = 1

Write the equation of the ellipse with center (-7, -8), a vertex at (-7, -22) and b = 7

((x + 7)^2)/(49) + ((y + 8)^2)/(196) = 1

Write the equation of the ellipse with vertices (-2, 1) and (-2, 7) and foci (-2, 2) and (-2, 6)

((x + 2)^2)/(5) + ((y - 4)^2)/(9) = 1

Write the equation of the ellipse with vertices (-12, 6) and (0, 6) and foci (-10, 6) and (-2, 6)

((x + 6)^2)/(36) + ((y - 6)^2)/(20) = 1

Write the equation of the ellipse with vertices (6, 2) and (6, 6) and foci (6, 3) and (6, 5)

((x - 6)^2)/(3) + ((y - 4)^2)/(4) = 1

Write the equation of the ellipse with vertices (-6, -1) and (14, -1) and foci (3, -1) and (5, -1)

((x - 4)^2)/(100) + ((y + 1)^2)/(99) = 1

Write the equation of the ellipse with vertices (1, -2) and (11, -2) and foci (2, -2) and (10, -2)

((x - 6)^2)/(25) + ((y + 2)^2)/(9) = 1

Write the equation of the ellipse with vertices (-4, -8) and (-4, 2) and foci (-4, -6) and (-4, 0)

((x + 4)^2)/(16) + ((y + 3)^2)/(25) = 1

Write the equation of the ellipse with vertices (-1, 6) and (3, 6) and foci (0, 6) and (2, 6)

((x - 1)^2)/(4) + ((y - 6)^2)/(3) = 1

Write the equation of the ellipse with vertices (-13, -4) and (7, -4) and foci (-9, -4) and (3, -4)

((x + 3)^2)/(100) + ((y + 4)^2)/(64) = 1

Write the equation of the ellipse with vertices (-11, 5) and (9, 5) and foci (-6, 5) and (4, 5)

((x + 1)^2)/(100) + ((y - 5)^2)/(75) = 1

Write the equation of the ellipse with major axis from (6, 2) to (6, -10) and minor axis from (11, -4) to (1, -4)

((x - 6)^2)/(25) + ((y + 4)^2)/(36) = 1

Write the equation of the ellipse with foci (1 + 2sqrt(2)2, 3/2) and (1 - 2sqrt(2)2, 3/2) and co-vertices (1/2, 11/2) and (1/2, -5/2)

(x - 1/2)^218 + (y - 3/2)^216 = 1

Worksheets · Precalculus

Equations of ellipses worksheet for Precalculus

Precalculus builds the ellipse from its parts. Each question gives the vertices and foci, or the center with a vertex or co-vertex and the other axis length. Before writing anything, decide whether the long axis is horizontal or vertical, because that decides which squared term gets a². When foci are given, use c² = a² - b².

Also taught in Algebra 2, at its own level and with its own examples.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Equations of Ellipses Date Period Write each equation. Write an ellipse from its center Write the equation of the ellipse with center $(5, - 4)$ , a vertex at $(- 5, - 4)$ and $b = 8$ Write the equation of the ellipse with center $(- 2, 8)$ , a co-vertex at $(- 2, 14)$ and $a = 11$ Write the equation of the ellipse with center $(- 6, - 1)$ , a vertex at $(2, - 1)$ and $b = 5$ Write the equation of the ellipse with center $(11, 1)$ , a co-vertex at $(4, 1)$ and $a = 12$ Write the equation of the ellipse with center $(1, - 4)$ , a vertex at $(1, 11)$ and $b = 14$ Write the equation of the ellipse with center $(10, - 2)$ , a co-vertex at $(10, - 9)$ and $a = 12$ Write the equation of the ellipse with center $(- 5, 8)$ , a vertex at $(- 5, 21)$ and $b = 6$ Write the equation of the ellipse with center $(5, - 11)$ , a vertex at $(5, - 23)$ and $b = 8$ Write the equation of the ellipse with center $(5, 8)$ , a vertex at $(5, - 2)$ and $b = 9$ Write the equation of the ellipse with center $(- 7, - 8)$ , a vertex at $(- 7, - 22)$ and $b = 7$ Write an ellipse from vertices and foci Write the equation of the ellipse with vertices $(- 2, 1)$ and $(- 2, 7)$ and foci $(- 2, 2)$ and $(- 2, 6)$ Write the equation of the ellipse with vertices $(- 12, 6)$ and $(0, 6)$ and foci $(- 10, 6)$ and $(- 2, 6)$ Write the equation of the ellipse with vertices $(6, 2)$ and $(6, 6)$ and foci $(6, 3)$ and $(6, 5)$ Write the equation of the ellipse with vertices $(- 6, - 1)$ and $(14, - 1)$ and foci $(3, - 1)$ and $(5, - 1)$ Write the equation of the ellipse with vertices $(1, - 2)$ and $(11, - 2)$ and foci $(2, - 2)$ and $(10, - 2)$ Write the equation of the ellipse with vertices $(- 4, - 8)$ and $(- 4, 2)$ and foci $(- 4, - 6)$ and $(- 4, 0)$ Write the equation of the ellipse with vertices $(4, - 10)$ and $(4, 10)$ and foci $(4, - 3)$ and $(4, 3)$ Write the equation of the ellipse with vertices $(- 1, 6)$ and $(3, 6)$ and foci $(0, 6)$ and $(2, 6)$ Write the equation of the ellipse with vertices $(- 13, - 4)$ and $(7, - 4)$ and foci $(- 9, - 4)$ and $(3, - 4)$ Write the equation of the ellipse with vertices $(- 11, 5)$ and $(9, 5)$ and foci $(- 6, 5)$ and $(4, 5)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Equations of Ellipses

Date Period

Write each equation.

Write an ellipse from its center
Write the equation of the ellipse with center $(5, - 4)$ , a vertex at $(- 5, - 4)$ and $b = 8$
Write the equation of the ellipse with center $(- 2, 8)$ , a co-vertex at $(- 2, 14)$ and $a = 11$
Write the equation of the ellipse with center $(- 6, - 1)$ , a vertex at $(2, - 1)$ and $b = 5$
Write the equation of the ellipse with center $(11, 1)$ , a co-vertex at $(4, 1)$ and $a = 12$
Write the equation of the ellipse with center $(1, - 4)$ , a vertex at $(1, 11)$ and $b = 14$
Write the equation of the ellipse with center $(10, - 2)$ , a co-vertex at $(10, - 9)$ and $a = 12$
Write the equation of the ellipse with center $(- 5, 8)$ , a vertex at $(- 5, 21)$ and $b = 6$
Write the equation of the ellipse with center $(5, - 11)$ , a vertex at $(5, - 23)$ and $b = 8$
Write the equation of the ellipse with center $(5, 8)$ , a vertex at $(5, - 2)$ and $b = 9$
Write the equation of the ellipse with center $(- 7, - 8)$ , a vertex at $(- 7, - 22)$ and $b = 7$

Write an ellipse from vertices and foci
Write the equation of the ellipse with vertices $(- 2, 1)$ and $(- 2, 7)$ and foci $(- 2, 2)$ and $(- 2, 6)$
Write the equation of the ellipse with vertices $(- 12, 6)$ and $(0, 6)$ and foci $(- 10, 6)$ and $(- 2, 6)$
Write the equation of the ellipse with vertices $(6, 2)$ and $(6, 6)$ and foci $(6, 3)$ and $(6, 5)$
Write the equation of the ellipse with vertices $(- 6, - 1)$ and $(14, - 1)$ and foci $(3, - 1)$ and $(5, - 1)$
Write the equation of the ellipse with vertices $(1, - 2)$ and $(11, - 2)$ and foci $(2, - 2)$ and $(10, - 2)$
Write the equation of the ellipse with vertices $(- 4, - 8)$ and $(- 4, 2)$ and foci $(- 4, - 6)$ and $(- 4, 0)$
Write the equation of the ellipse with vertices $(4, - 10)$ and $(4, 10)$ and foci $(4, - 3)$ and $(4, 3)$
Write the equation of the ellipse with vertices $(- 1, 6)$ and $(3, 6)$ and foci $(0, 6)$ and $(2, 6)$
Write the equation of the ellipse with vertices $(- 13, - 4)$ and $(7, - 4)$ and foci $(- 9, - 4)$ and $(3, - 4)$
Write the equation of the ellipse with vertices $(- 11, 5)$ and $(9, 5)$ and foci $(- 6, 5)$ and $(4, 5)$



Sigma Prep · sigmaprep.io/worksheets Name Equations of EllipsesAnswer keyVersion 2 Date Period Write each equation. Write an ellipse from its center Write the equation of the ellipse with center $(5, - 4)$ , a vertex at $(- 5, - 4)$ and $b = 8$ $\frac{{(x - 5)}^{2}}{100} + \frac{{(y + 4)}^{2}}{64} = 1$ Write the equation of the ellipse with center $(- 2, 8)$ , a co-vertex at $(- 2, 14)$ and $a = 11$ $\frac{{(x + 2)}^{2}}{121} + \frac{{(y - 8)}^{2}}{36} = 1$ Write the equation of the ellipse with center $(- 6, - 1)$ , a vertex at $(2, - 1)$ and $b = 5$ $\frac{{(x + 6)}^{2}}{64} + \frac{{(y + 1)}^{2}}{25} = 1$ Write the equation of the ellipse with center $(11, 1)$ , a co-vertex at $(4, 1)$ and $a = 12$ $\frac{{(x - 11)}^{2}}{49} + \frac{{(y - 1)}^{2}}{144} = 1$ Write the equation of the ellipse with center $(1, - 4)$ , a vertex at $(1, 11)$ and $b = 14$ $\frac{{(x - 1)}^{2}}{196} + \frac{{(y + 4)}^{2}}{225} = 1$ Write the equation of the ellipse with center $(10, - 2)$ , a co-vertex at $(10, - 9)$ and $a = 12$ $\frac{{(x - 10)}^{2}}{144} + \frac{{(y + 2)}^{2}}{49} = 1$ Write the equation of the ellipse with center $(- 5, 8)$ , a vertex at $(- 5, 21)$ and $b = 6$ $\frac{{(x + 5)}^{2}}{36} + \frac{{(y - 8)}^{2}}{169} = 1$ Write the equation of the ellipse with center $(5, - 11)$ , a vertex at $(5, - 23)$ and $b = 8$ $\frac{{(x - 5)}^{2}}{64} + \frac{{(y + 11)}^{2}}{144} = 1$ Write the equation of the ellipse with center $(5, 8)$ , a vertex at $(5, - 2)$ and $b = 9$ $\frac{{(x - 5)}^{2}}{81} + \frac{{(y - 8)}^{2}}{100} = 1$ Write the equation of the ellipse with center $(- 7, - 8)$ , a vertex at $(- 7, - 22)$ and $b = 7$ $\frac{{(x + 7)}^{2}}{49} + \frac{{(y + 8)}^{2}}{196} = 1$ Write an ellipse from vertices and foci Write the equation of the ellipse with vertices $(- 2, 1)$ and $(- 2, 7)$ and foci $(- 2, 2)$ and $(- 2, 6)$ $\frac{{(x + 2)}^{2}}{5} + \frac{{(y - 4)}^{2}}{9} = 1$ Write the equation of the ellipse with vertices $(- 12, 6)$ and $(0, 6)$ and foci $(- 10, 6)$ and $(- 2, 6)$ $\frac{{(x + 6)}^{2}}{36} + \frac{{(y - 6)}^{2}}{20} = 1$ Write the equation of the ellipse with vertices $(6, 2)$ and $(6, 6)$ and foci $(6, 3)$ and $(6, 5)$ $\frac{{(x - 6)}^{2}}{3} + \frac{{(y - 4)}^{2}}{4} = 1$ Write the equation of the ellipse with vertices $(- 6, - 1)$ and $(14, - 1)$ and foci $(3, - 1)$ and $(5, - 1)$ $\frac{{(x - 4)}^{2}}{100} + \frac{{(y + 1)}^{2}}{99} = 1$ Write the equation of the ellipse with vertices $(1, - 2)$ and $(11, - 2)$ and foci $(2, - 2)$ and $(10, - 2)$ $\frac{{(x - 6)}^{2}}{25} + \frac{{(y + 2)}^{2}}{9} = 1$ Write the equation of the ellipse with vertices $(- 4, - 8)$ and $(- 4, 2)$ and foci $(- 4, - 6)$ and $(- 4, 0)$ $\frac{{(x + 4)}^{2}}{16} + \frac{{(y + 3)}^{2}}{25} = 1$ Write the equation of the ellipse with vertices $(4, - 10)$ and $(4, 10)$ and foci $(4, - 3)$ and $(4, 3)$ $\frac{{(x - 4)}^{2}}{91} + \frac{y^{2}}{100} = 1$ Write the equation of the ellipse with vertices $(- 1, 6)$ and $(3, 6)$ and foci $(0, 6)$ and $(2, 6)$ $\frac{{(x - 1)}^{2}}{4} + \frac{{(y - 6)}^{2}}{3} = 1$ Write the equation of the ellipse with vertices $(- 13, - 4)$ and $(7, - 4)$ and foci $(- 9, - 4)$ and $(3, - 4)$ $\frac{{(x + 3)}^{2}}{100} + \frac{{(y + 4)}^{2}}{64} = 1$ Write the equation of the ellipse with vertices $(- 11, 5)$ and $(9, 5)$ and foci $(- 6, 5)$ and $(4, 5)$ $\frac{{(x + 1)}^{2}}{100} + \frac{{(y - 5)}^{2}}{75} = 1$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Equations of EllipsesAnswer keyVersion 2

Date Period

Write each equation.

Write an ellipse from its center
Write the equation of the ellipse with center $(5, - 4)$ , a vertex at $(- 5, - 4)$ and $b = 8$ $\frac{{(x - 5)}^{2}}{100} + \frac{{(y + 4)}^{2}}{64} = 1$
Write the equation of the ellipse with center $(- 2, 8)$ , a co-vertex at $(- 2, 14)$ and $a = 11$ $\frac{{(x + 2)}^{2}}{121} + \frac{{(y - 8)}^{2}}{36} = 1$
Write the equation of the ellipse with center $(- 6, - 1)$ , a vertex at $(2, - 1)$ and $b = 5$ $\frac{{(x + 6)}^{2}}{64} + \frac{{(y + 1)}^{2}}{25} = 1$
Write the equation of the ellipse with center $(11, 1)$ , a co-vertex at $(4, 1)$ and $a = 12$ $\frac{{(x - 11)}^{2}}{49} + \frac{{(y - 1)}^{2}}{144} = 1$
Write the equation of the ellipse with center $(1, - 4)$ , a vertex at $(1, 11)$ and $b = 14$ $\frac{{(x - 1)}^{2}}{196} + \frac{{(y + 4)}^{2}}{225} = 1$
Write the equation of the ellipse with center $(10, - 2)$ , a co-vertex at $(10, - 9)$ and $a = 12$ $\frac{{(x - 10)}^{2}}{144} + \frac{{(y + 2)}^{2}}{49} = 1$
Write the equation of the ellipse with center $(- 5, 8)$ , a vertex at $(- 5, 21)$ and $b = 6$ $\frac{{(x + 5)}^{2}}{36} + \frac{{(y - 8)}^{2}}{169} = 1$
Write the equation of the ellipse with center $(5, - 11)$ , a vertex at $(5, - 23)$ and $b = 8$ $\frac{{(x - 5)}^{2}}{64} + \frac{{(y + 11)}^{2}}{144} = 1$
Write the equation of the ellipse with center $(5, 8)$ , a vertex at $(5, - 2)$ and $b = 9$ $\frac{{(x - 5)}^{2}}{81} + \frac{{(y - 8)}^{2}}{100} = 1$
Write the equation of the ellipse with center $(- 7, - 8)$ , a vertex at $(- 7, - 22)$ and $b = 7$ $\frac{{(x + 7)}^{2}}{49} + \frac{{(y + 8)}^{2}}{196} = 1$

Write an ellipse from vertices and foci
Write the equation of the ellipse with vertices $(- 2, 1)$ and $(- 2, 7)$ and foci $(- 2, 2)$ and $(- 2, 6)$ $\frac{{(x + 2)}^{2}}{5} + \frac{{(y - 4)}^{2}}{9} = 1$
Write the equation of the ellipse with vertices $(- 12, 6)$ and $(0, 6)$ and foci $(- 10, 6)$ and $(- 2, 6)$ $\frac{{(x + 6)}^{2}}{36} + \frac{{(y - 6)}^{2}}{20} = 1$
Write the equation of the ellipse with vertices $(6, 2)$ and $(6, 6)$ and foci $(6, 3)$ and $(6, 5)$ $\frac{{(x - 6)}^{2}}{3} + \frac{{(y - 4)}^{2}}{4} = 1$
Write the equation of the ellipse with vertices $(- 6, - 1)$ and $(14, - 1)$ and foci $(3, - 1)$ and $(5, - 1)$ $\frac{{(x - 4)}^{2}}{100} + \frac{{(y + 1)}^{2}}{99} = 1$
Write the equation of the ellipse with vertices $(1, - 2)$ and $(11, - 2)$ and foci $(2, - 2)$ and $(10, - 2)$ $\frac{{(x - 6)}^{2}}{25} + \frac{{(y + 2)}^{2}}{9} = 1$
Write the equation of the ellipse with vertices $(- 4, - 8)$ and $(- 4, 2)$ and foci $(- 4, - 6)$ and $(- 4, 0)$ $\frac{{(x + 4)}^{2}}{16} + \frac{{(y + 3)}^{2}}{25} = 1$
Write the equation of the ellipse with vertices $(4, - 10)$ and $(4, 10)$ and foci $(4, - 3)$ and $(4, 3)$ $\frac{{(x - 4)}^{2}}{91} + \frac{y^{2}}{100} = 1$
Write the equation of the ellipse with vertices $(- 1, 6)$ and $(3, 6)$ and foci $(0, 6)$ and $(2, 6)$ $\frac{{(x - 1)}^{2}}{4} + \frac{{(y - 6)}^{2}}{3} = 1$
Write the equation of the ellipse with vertices $(- 13, - 4)$ and $(7, - 4)$ and foci $(- 9, - 4)$ and $(3, - 4)$ $\frac{{(x + 3)}^{2}}{100} + \frac{{(y + 4)}^{2}}{64} = 1$
Write the equation of the ellipse with vertices $(- 11, 5)$ and $(9, 5)$ and foci $(- 6, 5)$ and $(4, 5)$ $\frac{{(x + 1)}^{2}}{100} + \frac{{(y - 5)}^{2}}{75} = 1$

How to do these

Write the equation of the ellipse with vertices $(2, - 4)$ and $(2, 6)$ and foci $(2, - 3)$ and $(2, 5)$

The center is the midpoint, (2, 1). The vertices are 5 above and below it, so a = 5 and the long axis is vertical.
The foci are 4 away, so c = 4 and b² = 25 - 16 = 9.

The answer is $\frac{{(y - 1)}^{2}}{25} + \frac{{(x - 2)}^{2}}{9} = 1$ .

Where students go wrong

Using c² = a² + b² as for a hyperbola. In an ellipse the foci sit inside, so c is smaller than a and b² = a² - c².

Also called standard form of an ellipse or ellipse equation.

What you can put on this worksheet

Write the equation of an ellipse: Write the equation of the ellipse with center $(0, 1)$ , $a = 6$ , $b = 2$ and a vertical major axis → $\frac{x^{2}}{4} + \frac{{(y - 1)}^{2}}{36} = 1$
An ellipse from its vertices and foci: Write the equation of the ellipse with vertices $(- 5, 1)$ and $(5, 1)$ and foci $(- 1, 1)$ and $(1, 1)$ → $\frac{x^{2}}{25} + \frac{{(y - 1)}^{2}}{24} = 1$
From foci, axis lengths, eccentricity or a graph: Write the equation of the ellipse with major axis from $(6, 2)$ to $(6, - 10)$ and minor axis from $(11, - 4)$ to $(1, - 4)$ → $\frac{{(x - 6)}^{2}}{25} + \frac{{(y + 4)}^{2}}{36} = 1$
Foci and co-vertices, a half-integer center: Write the equation of the ellipse with foci $(\frac{1 + 2 \sqrt{2}}{2}, \frac{3}{2})$ and $(\frac{1 - 2 \sqrt{2}}{2}, \frac{3}{2})$ and co-vertices $(\frac{1}{2}, \frac{11}{2})$ and $(\frac{1}{2}, - \frac{5}{2})$ → $\frac{{(x - \frac{1}{2})}^{2}}{18} + \frac{{(y - \frac{3}{2})}^{2}}{16} = 1$

Questions about these worksheets

Yes. Every equations of ellipses sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the equations of ellipses questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are two levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 4: write the equation of an ellipse, an ellipse from its vertices and foci, from foci, axis lengths, eccentricity or a graph and foci and co-vertices, a half-integer center. Tick as many as you want and set how many of each, or let it spread them evenly.

Putting a and b underneath without squaring them. a = 5 gives 25, not 5.

Equations of ellipses worksheet for Precalculus

Select your difficulty

Write an ellipse from its center

Write an ellipse from vertices and foci

Write an ellipse from its center

Write an ellipse from vertices and foci

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these equations of ellipses worksheets free?+

Do the equations of ellipses worksheets come with an answer key?+

Are these equations of ellipses worksheets printable?+

What grade are these equations of ellipses worksheets for?+

Can I get a different version of the same equations of ellipses worksheet?+

Is this a free alternative to Kuta Software for equations of ellipses?+

Can I choose how hard the equations of ellipses questions are?+

What kinds of question are on a equations of ellipses worksheet?+

What do students get wrong with equations of ellipses?+