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: (x−h)2a2+(y−k)2b2=1\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 x2169+y2144=1\frac{x^2}{169}+\frac{y^2}{144}=1.

Example 2

easy
Length of major axis of x264+y236=1\frac{x^2}{64}+\frac{y^2}{36}=1?

Example 3

challenge
Find the equation of an ellipse with foci (0,±4)(0,\pm 4) passing through (3,0)(3,0).

Example 4

medium
Find the foci of x225+y216=1\frac{x^2}{25}+\frac{y^2}{16}=1.

Example 5

medium
An ellipse has vertices (0,±6)(0,\pm 6) and foci (0,±4)(0,\pm 4). Find its equation.

Example 6

medium
Convert 16x2+25y2=40016x^2 + 25y^2 = 400 to standard form.

Example 7

medium
Write the equation of an ellipse centered at the origin with foci at (±4,0)(\pm 4, 0) and a major axis of length 1010.

Example 8

challenge
An ellipse has foci (±3,0)(\pm 3, 0) and passes through (0,4)(0, 4). Find its equation.

Example 9

medium
Find the center of (x−1)216+(y−3)225=1\frac{(x-1)^2}{16}+\frac{(y-3)^2}{25}=1 and its foci.

Example 10

hard
Convert 4x2+9y2−16x+18y−11=04x^2 + 9y^2 - 16x + 18y - 11 = 0 to standard form.

Example 11

medium
Find the eccentricity of x2100+y264=1\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 x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1.

Example 13

medium
Find the foci of the ellipse x216+y225=1\frac{x^2}{16} + \frac{y^2}{25} = 1.

Example 14

medium
Why is c2=a2−b2c^2=a^2-b^2 (not a2+b2a^2+b^2) for an ellipse?

Example 15

hard
Does the point (2,1)(2, 1) lie inside, on, or outside the ellipse x29+y24=1\frac{x^2}{9}+\frac{y^2}{4}=1?

Example 16

easy
Find the vertices of x216+y24=1\frac{x^2}{16}+\frac{y^2}{4}=1.

Example 17

easy
In x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1, what are aa and bb?

Example 18

medium
An ellipse has a=10a=10 and e=25e=\frac{2}{5}. Find cc and bb.

Example 19

hard
Find bb for an ellipse with a=13a=13 and a focus at (5,0)(5, 0) (center at origin).

Example 20

medium
Find the foci of (x−2)2169+(y+1)2144=1\frac{(x-2)^2}{169}+\frac{(y+1)^2}{144}=1.
← Back to Ellipse Worked Examples Formula Explained