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 (8, 2), a vertex at (8, -10) and b = 6

((x - 8)^2)/(36) + ((y - 2)^2)/(144) = 1

Write the equation of the ellipse with center (-1, 3), a vertex at (-1, 18) and b = 7

((x + 1)^2)/(49) + ((y - 3)^2)/(225) = 1

Write the equation of the ellipse with center (8, 8), a vertex at (-3, 8) and b = 8

((x - 8)^2)/(121) + ((y - 8)^2)/(64) = 1

Write the equation of the ellipse with center (8, 10), a co-vertex at (8, 3) and a = 9

((x - 8)^2)/(81) + ((y - 10)^2)/(49) = 1

Write the equation of the ellipse with center (8, -1), a co-vertex at (1, -1) and a = 11

((x - 8)^2)/(49) + ((y + 1)^2)/(121) = 1

Write the equation of the ellipse with center (-8, -6), a co-vertex at (-8, 1) and a = 14

((x + 8)^2)/(196) + ((y + 6)^2)/(49) = 1

Write the equation of the ellipse with center (-5, -5), a co-vertex at (1, -5) and a = 14

((x + 5)^2)/(36) + ((y + 5)^2)/(196) = 1

Write the equation of the ellipse with center (4, 4), a vertex at (18, 4) and b = 6

((x - 4)^2)/(196) + ((y - 4)^2)/(36) = 1

Write the equation of the ellipse with center (-8, -3), a vertex at (-8, -12) and b = 5

((x + 8)^2)/(25) + ((y + 3)^2)/(81) = 1

Write the equation of the hyperbola with center (-3, 6), a vertex at (-3, 12) and b = 13

((y - 6)^2)/(36) - ((x + 3)^2)/(169) = 1

Write the equation of the hyperbola with center (10, -10), a vertex at (10, -16) and b = 7

((y + 10)^2)/(36) - ((x - 10)^2)/(49) = 1

Write the equation of the hyperbola with center (-10, 6), a vertex at (-10, -9) and b = 14

((y - 6)^2)/(225) - ((x + 10)^2)/(196) = 1

Write the equation of the hyperbola with center (-3, -11), a vertex at (-3, -26) and b = 8

((y + 11)^2)/(225) - ((x + 3)^2)/(64) = 1

Write the equation of the hyperbola with center (3, -5), a vertex at (3, 3) and b = 12

((y + 5)^2)/(64) - ((x - 3)^2)/(144) = 1

Write the equation of the hyperbola with center (-5, -7), a vertex at (9, -7) and b = 11

((x + 5)^2)/(196) - ((y + 7)^2)/(121) = 1

Write the equation of the hyperbola with center (-4, -4), a vertex at (-4, 9) and b = 15

((y + 4)^2)/(169) - ((x + 4)^2)/(225) = 1

Write the equation of the hyperbola with center (4, 2), a vertex at (17, 2) and b = 9

((x - 4)^2)/(169) - ((y - 2)^2)/(81) = 1

Write the equation of the hyperbola with center (3, 3), a vertex at (3, -2) and b = 10

((y - 3)^2)/(25) - ((x - 3)^2)/(100) = 1

Worksheets · Precalculus

Equations of ellipses and hyperbolas worksheet for Precalculus

Precalculus asks you to build these equations from parts: a center and a vertex with the length b, or, for some ellipses, a co-vertex with the length a. Read the distance from the center to the given point and use its direction to decide which axis is the long one, then remember that an ellipse adds the two terms while a hyperbola subtracts the second.

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

Not quite the one you need? Same words, different skill:

Standard Form of a Line (Algebra 1). Ax + By = C, with A positive and whole.
Graphing Lines: Standard Form (Algebra 1). Ax + By = C: rearrange it, or find both intercepts.
Writing Linear Equations (Algebra 1). Slope-intercept, point-slope and standard form.
Vertex Form of a Parabola (Algebra 1). y = a(x - h)² + k, both directions.
Equations of Circles (Precalculus). Center and radius, both directions.

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 Ellipses and Hyperbolas Date Period Give each equation in standard form. 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 $(8, 2)$ , a vertex at $(8, - 10)$ and $b = 6$ Write the equation of the ellipse with center $(- 1, 3)$ , a vertex at $(- 1, 18)$ and $b = 7$ Write the equation of the ellipse with center $(8, 8)$ , a vertex at $(- 3, 8)$ and $b = 8$ Write the equation of the ellipse with center $(8, 10)$ , a co-vertex at $(8, 3)$ and $a = 9$ Write the equation of the ellipse with center $(8, - 1)$ , a co-vertex at $(1, - 1)$ and $a = 11$ Write the equation of the ellipse with center $(- 8, - 6)$ , a co-vertex at $(- 8, 1)$ and $a = 14$ Write the equation of the ellipse with center $(- 5, - 5)$ , a co-vertex at $(1, - 5)$ and $a = 14$ Write the equation of the ellipse with center $(4, 4)$ , a vertex at $(18, 4)$ and $b = 6$ Write the equation of the ellipse with center $(- 8, - 3)$ , a vertex at $(- 8, - 12)$ and $b = 5$ Write a hyperbola from its center Write the equation of the hyperbola with center $(- 3, 6)$ , a vertex at $(- 3, 12)$ and $b = 13$ Write the equation of the hyperbola with center $(10, - 10)$ , a vertex at $(10, - 16)$ and $b = 7$ Write the equation of the hyperbola with center $(- 10, 6)$ , a vertex at $(- 10, - 9)$ and $b = 14$ Write the equation of the hyperbola with center $(- 3, - 11)$ , a vertex at $(- 3, - 26)$ and $b = 8$ Write the equation of the hyperbola with center $(3, - 5)$ , a vertex at $(3, 3)$ and $b = 12$ Write the equation of the hyperbola with center $(- 5, - 7)$ , a vertex at $(9, - 7)$ and $b = 11$ Write the equation of the hyperbola with center $(- 4, - 4)$ , a vertex at $(- 4, 9)$ and $b = 15$ Write the equation of the hyperbola with center $(4, 2)$ , a vertex at $(17, 2)$ and $b = 9$ Write the equation of the hyperbola with center $(0, 2)$ , a vertex at $(- 13, 2)$ and $b = 5$ Write the equation of the hyperbola with center $(3, 3)$ , a vertex at $(3, - 2)$ and $b = 10$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Ellipses and Hyperbolas

Date Period

Give each equation in standard form.

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 $(8, 2)$ , a vertex at $(8, - 10)$ and $b = 6$
Write the equation of the ellipse with center $(- 1, 3)$ , a vertex at $(- 1, 18)$ and $b = 7$
Write the equation of the ellipse with center $(8, 8)$ , a vertex at $(- 3, 8)$ and $b = 8$
Write the equation of the ellipse with center $(8, 10)$ , a co-vertex at $(8, 3)$ and $a = 9$
Write the equation of the ellipse with center $(8, - 1)$ , a co-vertex at $(1, - 1)$ and $a = 11$
Write the equation of the ellipse with center $(- 8, - 6)$ , a co-vertex at $(- 8, 1)$ and $a = 14$
Write the equation of the ellipse with center $(- 5, - 5)$ , a co-vertex at $(1, - 5)$ and $a = 14$
Write the equation of the ellipse with center $(4, 4)$ , a vertex at $(18, 4)$ and $b = 6$
Write the equation of the ellipse with center $(- 8, - 3)$ , a vertex at $(- 8, - 12)$ and $b = 5$

Write a hyperbola from its center
Write the equation of the hyperbola with center $(- 3, 6)$ , a vertex at $(- 3, 12)$ and $b = 13$
Write the equation of the hyperbola with center $(10, - 10)$ , a vertex at $(10, - 16)$ and $b = 7$
Write the equation of the hyperbola with center $(- 10, 6)$ , a vertex at $(- 10, - 9)$ and $b = 14$
Write the equation of the hyperbola with center $(- 3, - 11)$ , a vertex at $(- 3, - 26)$ and $b = 8$
Write the equation of the hyperbola with center $(3, - 5)$ , a vertex at $(3, 3)$ and $b = 12$
Write the equation of the hyperbola with center $(- 5, - 7)$ , a vertex at $(9, - 7)$ and $b = 11$
Write the equation of the hyperbola with center $(- 4, - 4)$ , a vertex at $(- 4, 9)$ and $b = 15$
Write the equation of the hyperbola with center $(4, 2)$ , a vertex at $(17, 2)$ and $b = 9$
Write the equation of the hyperbola with center $(0, 2)$ , a vertex at $(- 13, 2)$ and $b = 5$
Write the equation of the hyperbola with center $(3, 3)$ , a vertex at $(3, - 2)$ and $b = 10$



Sigma Prep · sigmaprep.io/worksheets Name Ellipses and HyperbolasAnswer keyVersion 2 Date Period Give each equation in standard form. 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 $(8, 2)$ , a vertex at $(8, - 10)$ and $b = 6$ $\frac{{(x - 8)}^{2}}{36} + \frac{{(y - 2)}^{2}}{144} = 1$ Write the equation of the ellipse with center $(- 1, 3)$ , a vertex at $(- 1, 18)$ and $b = 7$ $\frac{{(x + 1)}^{2}}{49} + \frac{{(y - 3)}^{2}}{225} = 1$ Write the equation of the ellipse with center $(8, 8)$ , a vertex at $(- 3, 8)$ and $b = 8$ $\frac{{(x - 8)}^{2}}{121} + \frac{{(y - 8)}^{2}}{64} = 1$ Write the equation of the ellipse with center $(8, 10)$ , a co-vertex at $(8, 3)$ and $a = 9$ $\frac{{(x - 8)}^{2}}{81} + \frac{{(y - 10)}^{2}}{49} = 1$ Write the equation of the ellipse with center $(8, - 1)$ , a co-vertex at $(1, - 1)$ and $a = 11$ $\frac{{(x - 8)}^{2}}{49} + \frac{{(y + 1)}^{2}}{121} = 1$ Write the equation of the ellipse with center $(- 8, - 6)$ , a co-vertex at $(- 8, 1)$ and $a = 14$ $\frac{{(x + 8)}^{2}}{196} + \frac{{(y + 6)}^{2}}{49} = 1$ Write the equation of the ellipse with center $(- 5, - 5)$ , a co-vertex at $(1, - 5)$ and $a = 14$ $\frac{{(x + 5)}^{2}}{36} + \frac{{(y + 5)}^{2}}{196} = 1$ Write the equation of the ellipse with center $(4, 4)$ , a vertex at $(18, 4)$ and $b = 6$ $\frac{{(x - 4)}^{2}}{196} + \frac{{(y - 4)}^{2}}{36} = 1$ Write the equation of the ellipse with center $(- 8, - 3)$ , a vertex at $(- 8, - 12)$ and $b = 5$ $\frac{{(x + 8)}^{2}}{25} + \frac{{(y + 3)}^{2}}{81} = 1$ Write a hyperbola from its center Write the equation of the hyperbola with center $(- 3, 6)$ , a vertex at $(- 3, 12)$ and $b = 13$ $\frac{{(y - 6)}^{2}}{36} - \frac{{(x + 3)}^{2}}{169} = 1$ Write the equation of the hyperbola with center $(10, - 10)$ , a vertex at $(10, - 16)$ and $b = 7$ $\frac{{(y + 10)}^{2}}{36} - \frac{{(x - 10)}^{2}}{49} = 1$ Write the equation of the hyperbola with center $(- 10, 6)$ , a vertex at $(- 10, - 9)$ and $b = 14$ $\frac{{(y - 6)}^{2}}{225} - \frac{{(x + 10)}^{2}}{196} = 1$ Write the equation of the hyperbola with center $(- 3, - 11)$ , a vertex at $(- 3, - 26)$ and $b = 8$ $\frac{{(y + 11)}^{2}}{225} - \frac{{(x + 3)}^{2}}{64} = 1$ Write the equation of the hyperbola with center $(3, - 5)$ , a vertex at $(3, 3)$ and $b = 12$ $\frac{{(y + 5)}^{2}}{64} - \frac{{(x - 3)}^{2}}{144} = 1$ Write the equation of the hyperbola with center $(- 5, - 7)$ , a vertex at $(9, - 7)$ and $b = 11$ $\frac{{(x + 5)}^{2}}{196} - \frac{{(y + 7)}^{2}}{121} = 1$ Write the equation of the hyperbola with center $(- 4, - 4)$ , a vertex at $(- 4, 9)$ and $b = 15$ $\frac{{(y + 4)}^{2}}{169} - \frac{{(x + 4)}^{2}}{225} = 1$ Write the equation of the hyperbola with center $(4, 2)$ , a vertex at $(17, 2)$ and $b = 9$ $\frac{{(x - 4)}^{2}}{169} - \frac{{(y - 2)}^{2}}{81} = 1$ Write the equation of the hyperbola with center $(0, 2)$ , a vertex at $(- 13, 2)$ and $b = 5$ $\frac{x^{2}}{169} - \frac{{(y - 2)}^{2}}{25} = 1$ Write the equation of the hyperbola with center $(3, 3)$ , a vertex at $(3, - 2)$ and $b = 10$ $\frac{{(y - 3)}^{2}}{25} - \frac{{(x - 3)}^{2}}{100} = 1$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Ellipses and HyperbolasAnswer keyVersion 2

Date Period

Give each equation in standard form.

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 $(8, 2)$ , a vertex at $(8, - 10)$ and $b = 6$ $\frac{{(x - 8)}^{2}}{36} + \frac{{(y - 2)}^{2}}{144} = 1$
Write the equation of the ellipse with center $(- 1, 3)$ , a vertex at $(- 1, 18)$ and $b = 7$ $\frac{{(x + 1)}^{2}}{49} + \frac{{(y - 3)}^{2}}{225} = 1$
Write the equation of the ellipse with center $(8, 8)$ , a vertex at $(- 3, 8)$ and $b = 8$ $\frac{{(x - 8)}^{2}}{121} + \frac{{(y - 8)}^{2}}{64} = 1$
Write the equation of the ellipse with center $(8, 10)$ , a co-vertex at $(8, 3)$ and $a = 9$ $\frac{{(x - 8)}^{2}}{81} + \frac{{(y - 10)}^{2}}{49} = 1$
Write the equation of the ellipse with center $(8, - 1)$ , a co-vertex at $(1, - 1)$ and $a = 11$ $\frac{{(x - 8)}^{2}}{49} + \frac{{(y + 1)}^{2}}{121} = 1$
Write the equation of the ellipse with center $(- 8, - 6)$ , a co-vertex at $(- 8, 1)$ and $a = 14$ $\frac{{(x + 8)}^{2}}{196} + \frac{{(y + 6)}^{2}}{49} = 1$
Write the equation of the ellipse with center $(- 5, - 5)$ , a co-vertex at $(1, - 5)$ and $a = 14$ $\frac{{(x + 5)}^{2}}{36} + \frac{{(y + 5)}^{2}}{196} = 1$
Write the equation of the ellipse with center $(4, 4)$ , a vertex at $(18, 4)$ and $b = 6$ $\frac{{(x - 4)}^{2}}{196} + \frac{{(y - 4)}^{2}}{36} = 1$
Write the equation of the ellipse with center $(- 8, - 3)$ , a vertex at $(- 8, - 12)$ and $b = 5$ $\frac{{(x + 8)}^{2}}{25} + \frac{{(y + 3)}^{2}}{81} = 1$

Write a hyperbola from its center
Write the equation of the hyperbola with center $(- 3, 6)$ , a vertex at $(- 3, 12)$ and $b = 13$ $\frac{{(y - 6)}^{2}}{36} - \frac{{(x + 3)}^{2}}{169} = 1$
Write the equation of the hyperbola with center $(10, - 10)$ , a vertex at $(10, - 16)$ and $b = 7$ $\frac{{(y + 10)}^{2}}{36} - \frac{{(x - 10)}^{2}}{49} = 1$
Write the equation of the hyperbola with center $(- 10, 6)$ , a vertex at $(- 10, - 9)$ and $b = 14$ $\frac{{(y - 6)}^{2}}{225} - \frac{{(x + 10)}^{2}}{196} = 1$
Write the equation of the hyperbola with center $(- 3, - 11)$ , a vertex at $(- 3, - 26)$ and $b = 8$ $\frac{{(y + 11)}^{2}}{225} - \frac{{(x + 3)}^{2}}{64} = 1$
Write the equation of the hyperbola with center $(3, - 5)$ , a vertex at $(3, 3)$ and $b = 12$ $\frac{{(y + 5)}^{2}}{64} - \frac{{(x - 3)}^{2}}{144} = 1$
Write the equation of the hyperbola with center $(- 5, - 7)$ , a vertex at $(9, - 7)$ and $b = 11$ $\frac{{(x + 5)}^{2}}{196} - \frac{{(y + 7)}^{2}}{121} = 1$
Write the equation of the hyperbola with center $(- 4, - 4)$ , a vertex at $(- 4, 9)$ and $b = 15$ $\frac{{(y + 4)}^{2}}{169} - \frac{{(x + 4)}^{2}}{225} = 1$
Write the equation of the hyperbola with center $(4, 2)$ , a vertex at $(17, 2)$ and $b = 9$ $\frac{{(x - 4)}^{2}}{169} - \frac{{(y - 2)}^{2}}{81} = 1$
Write the equation of the hyperbola with center $(0, 2)$ , a vertex at $(- 13, 2)$ and $b = 5$ $\frac{x^{2}}{169} - \frac{{(y - 2)}^{2}}{25} = 1$
Write the equation of the hyperbola with center $(3, 3)$ , a vertex at $(3, - 2)$ and $b = 10$ $\frac{{(y - 3)}^{2}}{25} - \frac{{(x - 3)}^{2}}{100} = 1$

How to do these

Write the equation of the hyperbola with center $(6, 1)$ , a vertex at $(10, 1)$ and $b = 3$

The vertex is 4 to the right of the center, so a = 4 and the x term comes first.
a² = 16 and b² = 9.
A hyperbola subtracts the second term.

The answer is $\frac{{(x - 6)}^{2}}{16} - \frac{{(y - 1)}^{2}}{9} = 1$ .

Where students go wrong

Putting a² under x when the vertices are stacked vertically. Vertices one above the other mean the main axis is vertical, so a² belongs under the (y - k)² term.

Also called conic equations.

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 → $\frac{x^{2}}{4} + \frac{{(y - 1)}^{2}}{36} = 1$
Write the equation of a hyperbola: Write the equation of the hyperbola with center $(0, 1)$ , $a = 6$ , $b = 2$ and a vertical transverse axis → $\frac{{(y - 1)}^{2}}{36} - \frac{x^{2}}{4} = 1$
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 $(1, 1)$ → $\frac{x^{2}}{25} + \frac{{(y - 1)}^{2}}{24} = 1$
A hyperbola from its vertices and foci: Write the equation of the hyperbola with vertices $(- 5, 1)$ and $(5, 1)$ and foci $(- 6, 1)$ and $(6, 1)$ → $\frac{x^{2}}{25} - \frac{{(y - 1)}^{2}}{11} = 1$

Questions about these worksheets

Yes. Every equations of ellipses and hyperbolas 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 and hyperbolas 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, write the equation of a hyperbola, an ellipse from its vertices and foci and a hyperbola from its vertices and foci. Tick as many as you want and set how many of each, or let it spread them evenly.

Putting a and b in without squaring them. The denominators are a squared and b squared, so a = 5 gives 25 underneath. The other one is losing track of which letter goes with which axis.

Equations of ellipses and hyperbolas worksheet for Precalculus

Select your difficulty

Write an ellipse from its center

Write a hyperbola from its center

Write an ellipse from its center

Write a hyperbola from its center

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

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