Ellipse Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Ellipse.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Pin a string at two foci and trace: the taut pencil draws an oval, an ellipse.

Common stuck point: The procedure for ellipse is the easy part; the trap is using $a^2+b^2$ for the foci. Asking "Are both squared terms positive, added, with different denominators equaling 1?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are both squared terms positive, added, with different denominators equaling 1?

Worked Examples

Example 1

easy

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

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

Answer

a = 5 \text{ (semi-major)}, \quad b = 3 \text{ (semi-minor)}

First step

The standard form is

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

where

a > b > 0

Full solution

2
Here $a^2 = 25$ and $b^2 = 9$ , so $a = 5$ and $b = 3$ .
3
The semi-major axis has length $a = 5$ (along the $x$ -axis) and the semi-minor axis has length $b = 3$ (along the $y$ -axis).

An ellipse

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

has semi-major axis

a

(the larger denominator's square root) and semi-minor axis

b

. The major axis lies along whichever variable has the larger denominator.

Example 2

medium

Find the foci of the ellipse

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

Example 3

medium

Why is

c^2=a^2-b^2

(not

a^2+b^2

) for an ellipse?

Example 4

hard

Show that as the eccentricity

e\to 0

, an ellipse approaches a circle.

Example 5

challenge

Find the perimeter of

\frac{x^2}{16}+\frac{y^2}{9}=1

using Ramanujan's approximation

P \approx \pi\left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

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 2

hard

Find the eccentricity of the ellipse

\frac{(x-2)^2}{36} + \frac{(y+1)^2}{20} = 1

and describe what it tells us about the shape.

Example 3

easy

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

, what are

a

and

b

Example 4

easy

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

wider or taller?

Example 5

easy

Give the center of

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

Example 6

easy

Find the lengths of the major and minor axes of

\frac{x^2}{36}+\frac{y^2}{16}=1

Example 7

easy

Write the equation of an ellipse centered at the origin with

a=4

(horizontal) and

b=2

Example 8

easy

What shape is

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

Example 9

easy

Find the vertices of

\frac{x^2}{49}+\frac{y^2}{4}=1

Example 10

easy

\frac{x^2}{4}+\frac{y^2}{25}=1

, which axis is major?

Example 11

medium

Find the foci of

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

Example 12

medium

Find the foci of

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

Example 13

medium

Find the eccentricity of

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

Example 14

medium

Write the equation of an ellipse with vertices

(\pm 5,0)

and foci

(\pm 3,0)

Example 15

medium

Find the center of

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

and its foci.

Example 16

medium

Convert

9x^2 + 4y^2 = 36

to standard ellipse form.

Example 17

medium

An ellipse has

a=10

and

e=\frac{2}{5}

. Find

c

and

b

Example 18

challenge

Find the equation of an ellipse with foci

(0,\pm 4)

passing through

(3,0)

Example 19

challenge

The sum of distances from a point on an ellipse to its foci is 10, with foci

(\pm 3,0)

. Find the equation.

Example 20

challenge

Find the eccentricity of

4x^2 + 9y^2 = 36

Example 21

medium

Find the co-vertices of

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

Example 22

medium

Find the foci of

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

Example 23

easy

State the semi-major and semi-minor axes of

\frac{x^2}{49} + \frac{y^2}{25}=1

Example 24

easy

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

taller or wider?

Example 25

easy

Write the equation of a horizontal ellipse centered at the origin with

a=6

and

b=2

Example 26

easy

Find the vertices of

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

Example 27

easy

How does

x^2 + y^2 = 25

relate to an ellipse?

Example 28

medium

Find the foci of

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

Example 29

medium

Find the foci of

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

Example 30

medium

Find the eccentricity of

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

Example 31

medium

Convert

16x^2 + 25y^2 = 400

to standard form.

Example 32

medium

Write the equation of an ellipse centered at

(2,-1)

with horizontal major axis length 10 and minor axis length 6.

Example 33

medium

Find the foci of

\frac{(x+1)^2}{25}+\frac{(y-3)^2}{9}=1

Example 34

medium

An ellipse has vertices

(0,\pm 6)

and foci

(0,\pm 4)

. Find its equation.

Example 35

medium

Area of the ellipse

\frac{x^2}{16}+\frac{y^2}{9}=1

Example 36

hard

Find the equation of the ellipse passing through

(0, 4)

and

(5, 0)

, centered at the origin with axes on the coordinate axes.

Example 37

hard

Sum of distances from any point on

\frac{x^2}{49}+\frac{y^2}{24}=1

to the foci?

Example 38

hard

Convert

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

to standard form.

Example 39

hard

Find

b

for an ellipse with

a=13

and a focus at

(5, 0)

(center at origin).

Example 40

hard

Does the point

(2, 1)

lie inside, on, or outside the ellipse

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

Example 41

challenge

An ellipse has foci

(\pm 3, 0)

and passes through

(0, 4)

. Find its equation.

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Related Concepts

Equation of a Circle Conic Sections Overview Hyperbola

Background Knowledge

These ideas may be useful before you work through the harder examples.

equation of circle