Pre-Calculus Conic sections

Ellipses (standard form, foci, center at (h, k))

Q: How do you find the foci of an ellipse?

Compute c equals the square root of a squared minus b squared; the foci lie a distance c from the center along the major axis.

Q: What is the equation of a shifted ellipse?

x minus h squared over a squared plus y minus k squared over b squared equals 1, centered at h, k.

20 practice questions 0 video lessons Theory + worked examples

Ellipses

Texas Precalculus (TEKS) • Standard P.3(H) • Conic Sections

Ellipses is a topic in Conic Sections in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.3(H), which requires students to write the equation of an ellipse with center (h, k).

An ellipse is the set of points whose distances to two foci add to a constant, with standard form \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\).

Practice 20 questions

Practice questions

Every question with a fully worked solution.

Start practising

Theory

An ellipse is the set of points whose distances to two fixed foci add to a constant. Centered at the origin its standard form is

\[\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1,\qquad a>b.\]

The major axis (length \(2a\)) lies along the variable with the larger denominator; the minor axis has length \(2b\). The foci sit a distance \(c\) from the center, where

\[c^2=a^2-b^2.\]

The larger denominator points to the major axis. If it is under \(y^2\), the ellipse is taller than it is wide.

An ellipse: the sum of distances to the two foci is constant.

Standard form and the foci.

Standard form and the focal distance:

\[\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1,\qquad c^2=a^2-b^2\]

x squared over a squared plus y squared over b squared equals 1; c squared equals a squared minus b squared

Center \((h,k)\): replace \(x,y\) with \(x-h,\ y-k\).

How to analyze an ellipse

Identify \(a^2\) and \(b^2\) (\(a^2\) is the larger).
Major axis lies along the larger-denominator variable, length \(2a\).
Foci: \(c=\sqrt{a^2-b^2}\) from the center, along the major axis.
Center from the \((x-h),(y-k)\) shifts.

Example 1 — Axes and foci

For \(\dfrac{x^2}{25}+\dfrac{y^2}{9}=1\), find the axes lengths and foci.

Solution

\(a^2=25,\ b^2=9\), so \(a=5,\ b=3\); \(c^2=25-9=16\).

\(\text{major axis}\)	\(=\)	\(2a=10\)
\(\text{minor axis}\)	\(=\)	\(2b=6\)
\(\text{foci}\)	\(=\)	\((\pm 4,0)\)

major axis 10, minor axis 6, foci at plus or minus 4

Example 2 — Vertical major axis

Describe \(\dfrac{x^2}{9}+\dfrac{y^2}{25}=1\).

Solution

Here the larger denominator is under \(y^2\), so the major axis is vertical.

\(a=5\ (\text{on }y),\ b=3\)
\(\text{foci}\)	\(=\)	\((0,\pm 4)\)

major axis vertical, foci at 0 comma plus or minus 4

Example 3 — Write an equation

Write the ellipse centered at the origin with \(a=6\) horizontal and \(b=4\).

Solution

Place \(a^2\) under \(x^2\).

\[\dfrac{x^2}{36}+\dfrac{y^2}{16}=1\]

equation is x squared over 36 plus y squared over 16 equals 1

Example 4 — Center at (h, k)

Give the center of \(\dfrac{(x-2)^2}{9}+\dfrac{(y+1)^2}{4}=1\).

Solution

Read the shifts directly.

\(\text{center}\)

\(=\)

\((2,-1)\)

center is 2 comma negative 1

Common pitfalls

\(a^2\) is the larger denominator, and it decides the major axis's direction.

Foci use \(c^2=a^2-b^2\) (subtract), unlike the hyperbola.

The foci lie on the major axis, not the minor one.

Frequently asked questions

What is an ellipse?

The set of points whose distances to two foci sum to a constant; its standard form is \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\).

How do you find the foci of an ellipse?

Compute \(c=\sqrt{a^2-b^2}\); the foci lie a distance \(c\) from the center along the major axis.

How do you tell which axis is major?

The larger denominator marks the major axis. If it is under \(y^2\), the ellipse is taller; under \(x^2\), it is wider.

What is the equation of a shifted ellipse?

\(\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1\), centered at \((h,k)\).

← Previous subtopic

Parabolas (standard form, focus, directrix, axis)

Next subtopic →

Hyperbolas (standard form, foci, asymptotes)

Ellipses (standard form, foci, center at (h, k))

Ellipses

📖 Theory

How to analyze an ellipse

Common pitfalls

Frequently asked questions

Theory