Practice Ellipse in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

The set of all points in a plane where the sum of the distances to two fixed points (foci) is constant. Standard form: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ .

Imagine pinning two ends of a loose string to a board (these are the foci), then tracing a curve with a pencil keeping the string taut. The resulting oval shape is an ellipse. A circle is just a special ellipse where both foci coincide.

Showing a random 20 of 50 problems.

Example 1

medium

Find the foci of

\frac{x^2}{169}+\frac{y^2}{144}=1

Example 2

easy

Length of major axis of

\frac{x^2}{64}+\frac{y^2}{36}=1

Example 3

challenge

Find the equation of an ellipse with foci

(0,\pm 4)

passing through

(3,0)

Example 4

medium

Find the foci of

\frac{x^2}{25}+\frac{y^2}{16}=1

Example 5

medium

An ellipse has vertices

(0,\pm 6)

and foci

(0,\pm 4)

. Find its equation.

Example 6

medium

Convert

16x^2 + 25y^2 = 400

to standard form.

Example 7

medium

Write the equation of an ellipse centered at the origin with foci at

(\pm 4, 0)

and a major axis of length

10

Example 8

challenge

An ellipse has foci

(\pm 3, 0)

and passes through

(0, 4)

. Find its equation.

Example 9

medium

Find the center of

\frac{(x-1)^2}{16}+\frac{(y-3)^2}{25}=1

and its foci.

Example 10

hard

Convert

4x^2 + 9y^2 - 16x + 18y - 11 = 0

to standard form.

Example 11

medium

Find the eccentricity of

\frac{x^2}{100}+\frac{y^2}{64}=1

Example 12

easy

Find the lengths of the semi-major and semi-minor axes of the ellipse

\frac{x^2}{25} + \frac{y^2}{9} = 1

Example 13

medium

Find the foci of the ellipse

\frac{x^2}{16} + \frac{y^2}{25} = 1

Example 14

medium

Why is

c^2=a^2-b^2

(not

a^2+b^2

) for an ellipse?

Example 15

hard

Does the point

(2, 1)

lie inside, on, or outside the ellipse

\frac{x^2}{9}+\frac{y^2}{4}=1

Example 16

easy

Find the vertices of

\frac{x^2}{16}+\frac{y^2}{4}=1

Example 17

easy

\frac{x^2}{25} + \frac{y^2}{9} = 1

, what are

a

and

b

Example 18

medium

An ellipse has

a=10

and

e=\frac{2}{5}

. Find

c

and

b

Example 19

hard

Find

b

for an ellipse with

a=13

and a focus at

(5, 0)

(center at origin).

Example 20

medium

Find the foci of

\frac{(x-2)^2}{169}+\frac{(y+1)^2}{144}=1

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Related Concepts

Equation of a Circle Conic Sections Overview Hyperbola