Ellipses in Standard Form

Learn to write, graph, and analyze ellipses using the standard form equation.

Advanced25 minLesson

Definition

An ellipse is the set of all points

(x, y)

such that the sum of the distances from two fixed points (called foci) is constant.

Standard Form Equations

Horizontal ellipse (wider than tall):

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \quad \text{where } a > b

Vertical ellipse (taller than wide):

\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \quad \text{where } a > b

Key Components

Center: $(h, k)$
Semi-major axis: $a$ (always the larger value)
Semi-minor axis: $b$ (always the smaller value)
Relationship: $c^2 = a^2 - b^2$ where $c$ is the distance from center to each focus

Try it now

What is the center of the ellipse

\frac{(x-3)^2}{16} + \frac{(y+2)^2}{9} = 1

Worked Examples

Find the center, vertices, co-vertices, and foci of the ellipse: $\frac{(x-2)^2}{25} + \frac{(y+3)^2}{9} = 1$

Identify the center $(h, k)$

From $(x-2)^2$ and $(y+3)^2$ , we have $h = 2$ and $k = -3$ → Center: $(2, -3)$

Find $a^2$ and $b^2$

$a^2 = 25$ (larger), $b^2 = 9$ (smaller) → $a = 5$ , $b = 3$

Determine orientation

Since $a^2 = 25$ is under $(x-h)^2$ , the major axis is horizontal → Horizontal ellipse

Find vertices (on major axis)

$(h \pm a, k) = (2 \pm 5, -3)$ → Vertices: $(-3, -3)$ and $(7, -3)$

Find co-vertices (on minor axis)

$(h, k \pm b) = (2, -3 \pm 3)$ → Co-vertices: $(2, -6)$ and $(2, 0)$

Calculate $c$ for foci

$c^2 = a^2 - b^2 = 25 - 9 = 16$ , so $c = 4$

Find foci (on major axis)

$(h \pm c, k) = (2 \pm 4, -3)$ → Foci: $(-2, -3)$ and $(6, -3)$

Answer: Center: $(2, -3)$ ; Vertices: $(-3, -3)$ , $(7, -3)$ ; Co-vertices: $(2, -6)$ , $(2, 0)$ ; Foci: $(-2, -3)$ , $(6, -3)$

Common Mistakes

Confusing which denominator gives $a^2$ vs $b^2$

Why it's wrong: Students assume $a^2$ is always under $x^2$ , but $a > b$ by definition, so $a^2$ is the larger denominator.

Correct: Always identify: $a^2$ = larger denominator, $b^2$ = smaller denominator. Then determine orientation based on which variable has $a^2$ .

Using $c^2 = a^2 + b^2$ (hyperbola formula) instead of $c^2 = a^2 - b^2$

Why it's wrong: Hyperbolas use addition, ellipses use subtraction. This is a critical difference.

Correct: For ellipses: $c^2 = a^2 - b^2$ . Remember: foci are inside the ellipse, so $c < a$ .

Forgetting to change signs when reading center from equation

Why it's wrong: The equation has $(x - h)^2$ , so $(x + 3)^2$ means $h = -3$ , not $h = 3$ .

Correct: $(x - h)^2$ with $(x + 3)^2$ means $-h = 3$ , so $h = -3$ .

Placing foci on the minor axis

Why it's wrong: Students may forget that foci are always on the major axis (the longer one).

Correct: Foci are always on the major axis, at distance $c$ from the center.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Ellipses Explained: Vertex, Foci, Eccentricity, and Equations
by Understand The Math
Watch on YouTube

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Practice Problems

17 problems

Problem 1 of 17

Easy

What is the center of the ellipse $\frac{(x-3)^2}{16} + \frac{(y+2)^2}{9} = 1$ ?

Why It Matters

Ellipses appear throughout science and engineering:

Astronomy: Planets orbit the Sun in elliptical paths (Kepler's First Law)
Architecture: Whispering galleries use elliptical ceilings for acoustic effects
Medicine: Lithotripsy machines use elliptical reflectors to focus sound waves on kidney stones
Optics: Elliptical mirrors reflect all light from one focus to the other

Understanding the standard form helps us analyze and design these systems mathematically.

Real World Applications

Planetary Orbits

All planets orbit the Sun in elliptical paths with the Sun at one focus.

Example:

Earth's orbit has $a \approx 149.6$ million km and eccentricity $e \approx 0.017$ , making it nearly circular.

1Try It Yourself

A comet orbits the Sun in an ellipse with semi-major axis $a = 18$ AU and $c = 17$ AU (distance from center to Sun).

Find the semi-minor axis $b$ .

Step 1: Write the mathematical expression

Use $c^2 = a^2 - b^2$ :

Whispering Galleries

In an elliptical room, sound from one focus reflects to the other focus, allowing whispers to travel across the room.

Example:

The Capitol Building in Washington D.C. has an elliptical room where this effect occurs.

2Try It Yourself

An elliptical hall is 30 meters long and 20 meters wide.

How far apart are the two focal points (whispering spots)?

Step 1: Write the mathematical expression

Find $2c$ where $a = 15$ and $b = 10$ :

Key Takeaways

1An ellipse in standard form is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ (horizontal) or with $a^2$ and $b^2$ swapped (vertical)
2The center is at $(h, k)$ , and $a > b$ always (larger denominator determines orientation)
3Vertices are on the major axis at distance $a$ from center; co-vertices are on minor axis at distance $b$
4Foci are on the major axis at distance $c$ from center, where $c^2 = a^2 - b^2$
5To convert general form to standard form, complete the square for both variables

Frequently Asked Questions

Look at which variable has the larger denominator. If

a^2

(larger) is under

(x-h)^2

, the major axis is horizontal. If

a^2

is under

(y-k)^2

, the major axis is vertical.

Glossary

Ellipse: The set of all points where the sum of distances to two foci is constant
Semi-major axis: Half the length of the longest diameter, denoted $a$
Semi-minor axis: Half the length of the shortest diameter, denoted $b$
Focus (pl. foci): One of two fixed points inside the ellipse; $c^2 = a^2 - b^2$
Vertex: An endpoint of the major axis
Co-vertex: An endpoint of the minor axis
Eccentricity: The ratio $e = c/a$ measuring how elongated the ellipse is (0 = circle, close to 1 = very elongated)

Formula Card

Standard Form (horizontal)

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1

$a > b$, major axis horizontal

Standard Form (vertical)

\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1

$a > b$, major axis vertical

Relationship

c^2 = a^2 - b^2

$c$ = distance from center to focus

Eccentricity

e = \frac{c}{a}

$0 < e < 1$ for all ellipses