Lesson 12.1: The Ellipse

New Concept

An ellipse is a curve defined by two focal points. In this lesson, you'll master its standard equations, enabling you to graph ellipses centered at the origin or elsewhere and solve real-world problems like the acoustics of whispering chambers.

What’s next

Next up, you'll tackle practice cards on writing ellipse equations. Then, we'll dive into interactive examples to master graphing them.

Property

An ellipse is the set of all points $(x, y)$ in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci). Every ellipse has two axes of symmetry. The longer axis is called the major axis, and the shorter axis is called the minor axis. Each endpoint of the major axis is the vertex of the ellipse, and each endpoint of the minor axis is a co-vertex. The center of an ellipse is the midpoint of both the major and minor axes.

Examples

• If the foci of an ellipse are at $(-4, 0)$ and $(4, 0)$ and the constant sum of the distances is 12, then any point $(x,y)$ on the ellipse satisfies the equation $\sqrt{(x+4)^2 + y^2} + \sqrt{(x-4)^2 + y^2} = 12$.
• An ellipse with vertices at $(0, \pm 6)$ and co-vertices at $(\pm 3, 0)$ has a vertical major axis of length $2(6)=12$ and a horizontal minor axis of length $2(3)=6$. Its center is at the origin $(0,0)$.
• The foci of an ellipse always lie on the major axis. If an ellipse has vertices at $(\pm 9, 0)$, its foci must be located at $(\pm c, 0)$ where $c < 9$.

Explanation

Think of an ellipse as a stretched circle defined by two anchor points (foci). The sum of the distances from any point on the curve to these two foci is always the same, creating its unique oval shape.

Property

The standard form of the equation of an ellipse with center $(0, 0)$ and major axis on the x-axis is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b$, the length of the major axis is $2a$, the vertices are $(\pm a, 0)$, the co-vertices are $(0, \pm b)$, and the foci are $(\pm c, 0)$, where $c^2 = a^2 - b^2$. The standard form of the equation of an ellipse with center $(0, 0)$ and major axis on the y-axis is $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ where $a > b$, the length of the major axis is $2a$, the vertices are $(0, \pm a)$, the co-vertices are $(\pm b, 0)$, and the foci are $(0, \pm c)$, where $c^2 = a^2 - b^2$.

Examples

• An ellipse has vertices at $(\pm 7, 0)$ and foci at $(\pm 5, 0)$. Here, $a=7$ and $c=5$. We find $b^2$ using $c^2 = a^2 - b^2$, so $25 = 49 - b^2$, which gives $b^2=24$. The equation is $\frac{x^2}{49} + \frac{y^2}{24} = 1$.
• For the ellipse $\frac{x^2}{16} + \frac{y^2}{64} = 1$, we see $a^2=64$ and $b^2=16$. Since $a^2$ is under $y^2$, the major axis is vertical. The vertices are $(0, \pm 8)$ and the co-vertices are $(\pm 4, 0)$.
• To find the foci of $\frac{x^2}{81} + \frac{y^2}{45} = 1$, we have $a^2=81$ and $b^2=45$. Then $c^2 = 81 - 45 = 36$, so $c=6$. Since the major axis is horizontal, the foci are at $(\pm 6, 0)$.

Explanation

The larger denominator is always $a^2$ and it reveals the major axis. If it's under $x^2$, the ellipse is horizontal ('wide'). If it's under $y^2$, the ellipse is vertical ('tall'). The value $c$ is the distance from the center to a focus.

Property

The standard form of the equation of an ellipse with center $(h, k)$ and major axis parallel to the $x$-axis is $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$ where $a > b$. The vertices are $(h \pm a, k)$ and the foci are $(h \pm c, k)$. The standard form of the equation of an ellipse with center $(h, k)$ and major axis parallel to the $y$-axis is $\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$ where $a > b$. The vertices are $(h, k \pm a)$ and the foci are $(h, k \pm c)$. For both forms, $c^2 = a^2 - b^2$.

Examples

• The ellipse $\frac{(x-2)^2}{25} + \frac{(y+4)^2}{16} = 1$ is centered at $(2, -4)$. Since $a^2=25$ is under the $x$ term, it's a horizontal ellipse with vertices at $(2 \pm 5, -4)$, which are $(7, -4)$ and $(-3, -4)$.
• An ellipse has vertices at $(3, 2)$ and $(3, 12)$ and a focus at $(3, 7+\sqrt{11})$. The center is $(3, 7)$. The major axis is vertical with $a=5$ ($a^2=25$) and $c=\sqrt{11}$ ($c^2=11$). Thus, $b^2 = 25 - 11 = 14$. The equation is $\frac{(x-3)^2}{14} + \frac{(y-7)^2}{25} = 1$.
• Find the foci of $\frac{(x+1)^2}{9} + \frac{(y-3)^2}{34} = 1$. The center is $(-1, 3)$ and the major axis is vertical. $a^2=34$ and $b^2=9$. Then $c^2 = 34-9=25$, so $c=5$. The foci are at $(h, k \pm c)$, which are $(-1, 3 \pm 5)$, or $(-1, 8)$ and $(-1, -2)$.

Explanation

This form simply shifts the center of the ellipse from $(0,0)$ to a new point $(h, k)$. All other properties, including the shape and size defined by $a$ and $b$, move along with the center without changing.

Property

To convert an ellipse from the general form $ax^2 + by^2 + cx + dy + e = 0$ to standard form: 1. Group terms with the same variable and move the constant to the opposite side. 2. Factor out the coefficients of the $x^2$ and $y^2$ terms. 3. Complete the square for each variable, adding the balanced values to the other side. 4. Rewrite as perfect squares. 5. Divide both sides by the constant term to make the equation equal 1.

Examples

• Convert $4x^2 + 25y^2 + 8x - 100y + 4 = 0$. Grouping and factoring gives $4(x^2+2x) + 25(y^2-4y) = -4$. Completing the square yields $4(x+1)^2 + 25(y-2)^2 = -4 + 4(1) + 25(4) = 100$. Divide by 100 to get $\frac{(x+1)^2}{25} + \frac{(y-2)^2}{4} = 1$.
• The equation $x^2 + 9y^2 - 4x + 54y + 49 = 0$ becomes $(x^2-4x) + 9(y^2+6y) = -49$. Completing the square gives $(x-2)^2 + 9(y+3)^2 = -49 + 4 + 9(9) = 36$. The standard form is $\frac{(x-2)^2}{36} + \frac{(y+3)^2}{4} = 1$.
• Rewrite $16x^2 + y^2 - 128x + 240 = 0$. This is $16(x^2-8x) + y^2 = -240$. Completing the square gives $16(x-4)^2 + y^2 = -240 + 16(16) = 16$. The standard form is $\frac{(x-4)^2}{1} + \frac{y^2}{16} = 1$.

Explanation

The general form hides an ellipse's properties. Completing the square for both $x$ and $y$ reorganizes the equation to reveal the center $(h, k)$, axis lengths $a$ and $b$, and orientation, making it easy to analyze.