Lesson 12.2: The Hyperbola

New Concept

A hyperbola is a set of points where the difference in distances to two foci is constant. We'll explore its unique shape, identify key features like vertices and asymptotes, and master writing and graphing its equations.

What’s next

First, we'll break down the formal definition. Then, you'll tackle interactive examples to find vertices, foci, and graph hyperbolas yourself.

Property

In analytic geometry, a hyperbola is a conic section formed by intersecting a right circular cone with a plane at an angle such that both halves of the cone are intersected. A hyperbola is also the set of all points $(x, y)$ in a plane such that the difference of the distances between $(x, y)$ and the foci is a positive constant. The transverse axis is a line segment that passes through the center of the hyperbola and has vertices as its endpoints. The conjugate axis is perpendicular to the transverse axis. The center of a hyperbola is the midpoint of both the transverse and conjugate axes. Every hyperbola also has two asymptotes that pass through its center.

Examples

• A point $(x, y)$ lies on a hyperbola with foci at $(-5, 0)$ and $(5, 0)$. For this hyperbola, the difference of the distances from $(x, y)$ to the foci is always 6.
• If a hyperbola has vertices at $(0, ±3)$, its transverse axis is the line segment on the y-axis between them, with a length of $2a=6$.
• The central rectangle is a construction tool whose corners define the slope of the asymptotes, which guide the shape of the hyperbola's branches.

Explanation

A hyperbola has two separate, mirror-image branches. For any point on a branch, its distance to one focus minus its distance to the other focus is a constant value. Asymptotes are straight lines that the branches approach but never touch.

Property

The standard form of the equation of a hyperbola with center $(0, 0)$ and transverse axis on the x-axis is

where the vertices are $(± a, 0)$ and the foci are $(± c, 0)$.
The standard form with a transverse axis on the y-axis is

where the vertices are $(0, ± a)$ and the foci are $(0, ± c)$.
For both cases, the relationship between $a, b,$ and $c$ is $c^2 = a^2 + b^2$. The equations of the asymptotes are $y = ± \frac{b}{a} x$ for a horizontal hyperbola and $y = ± \frac{a}{b} x$ for a vertical hyperbola.

Examples

• Given the equation $\frac{x^2}{9} - \frac{y^2}{16} = 1$, the hyperbola is horizontal. Here, $a^2=9$ and $b^2=16$. The vertices are $(± 3, 0)$. We find $c$ with $c = \sqrt{9+16} = 5$, so the foci are $(± 5, 0)$.
• For a hyperbola with vertices $(0, ± 2)$ and foci $(0, ± 2\sqrt{5})$, the axis is vertical. Here $a=2$ and $c=2\sqrt{5}$. We find $b^2 = c^2 - a^2 = (2\sqrt{5})^2 - 2^2 = 20 - 4 = 16$. The equation is $\frac{y^2}{4} - \frac{x^2}{16} = 1$.
• The asymptotes for the hyperbola $\frac{y^2}{64} - \frac{x^2}{36} = 1$ are given by $y = ± \frac{a}{b}x = ± \frac{8}{6}x = ± \frac{4}{3}x$.

Explanation

For a hyperbola centered at the origin, the variable with the positive coefficient ($x^2$ or $y^2$) tells you if it opens horizontally or vertically. The value $a^2$ is always under the positive term. The relationship $c^2 = a^2 + b^2$ helps you find the foci.

Property

The standard form of the equation of a hyperbola with center $(h, k)$ and transverse axis parallel to the x-axis is

where the vertices are $(h ± a, k)$ and the foci are $(h ± c, k)$.
The standard form with a transverse axis parallel to the y-axis is

where the vertices are $(h, k ± a)$ and the foci are $(h, k ± c)$.
For both forms, $c^2 = a^2 + b^2$. The asymptotes are given by $y = ± \frac{b}{a}(x-h) + k$ for a horizontal hyperbola and $y = ± \frac{a}{b}(x-h) + k$ for a vertical one.

Examples

• The equation $\frac{(x-1)^2}{9} - \frac{(y-2)^2}{16} = 1$ describes a hyperbola with center $(1, 2)$, a horizontal axis, and vertices at $(1 ± 3, 2)$, which are $(-2, 2)$ and $(4, 2)$.
• Find the equation of a hyperbola with vertices at $(0, -2)$ and $(6, -2)$ and foci at $(-2, -2)$ and $(8, -2)$. The center is the midpoint $(3, -2)$. The axis is horizontal. $2a = |6-0|=6 \implies a=3$. $2c = |8 - (-2)| = 10 \implies c=5$. Then $b^2 = 5^2 - 3^2 = 16$. The equation is $\frac{(x-3)^2}{9} - \frac{(y+2)^2}{16} = 1$.
• For the hyperbola $\frac{(y+3)^2}{100} - \frac{(x-3)^2}{64} = 1$, the center is $(3, -3)$, $a=10$, $b=8$. The asymptotes are $y+3 = ± \frac{10}{8}(x-3)$, which simplifies to $y+3 = ± \frac{5}{4}(x-3)$.

Explanation

To shift a hyperbola, we move its center from the origin to a new point $(h, k)$. The standard formulas are updated by replacing $x$ with $(x-h)$ and $y$ with $(y-k)$ to reflect this translation.

Property

To sketch the graph of a hyperbola from its standard form equation:

1. Convert the general form to standard form if necessary. Identify the center $(h, k)$, $a$, and $b$.
2. Determine the orientation (horizontal or vertical) from the positive term.
3. Plot the center, vertices (e.g., $(h ± a, k)$), and co-vertices (e.g., $(h, k ± b)$).
4. Sketch the central rectangle, which has sides that pass through the vertices and co-vertices.
5. Draw the asymptotes as extended diagonals of the central rectangle.
6. Draw the two hyperbolic branches starting from each vertex and approaching the asymptotes.

Examples

• To graph $\frac{x^2}{49} - \frac{y^2}{16} = 1$, note the center is $(0,0)$ and it's horizontal. Vertices are $(± 7, 0)$, co-vertices are $(0, ± 4)$. Sketch the central rectangle and its diagonal asymptotes, then draw the branches from the vertices.
• To graph $9x^2 - 4y^2 - 36x - 40y - 388 = 0$, first complete the square to get the standard form $\frac{(x - 2)^2}{36} - \frac{(y + 5)^2}{81} = 1$. The center is $(2, -5)$ and the transverse axis is horizontal.
• Graph the hyperbola $\frac{(y-3)^2}{9} - \frac{(x-3)^2}{9} = 1$. The center is $(3,3)$. It is a vertical hyperbola with $a=3$ and $b=3$. The vertices are at $(3, 3 ± 3)$, or $(3, 6)$ and $(3, 0)$.

Explanation

To graph a hyperbola, find its center, vertices, and co-vertices. These points define a 'central rectangle.' The diagonals of this box become the asymptotes, which act as perfect guides for sketching the hyperbola's two curved branches.

Property

Hyperbolas can model real-world phenomena like the shape of cooling towers. To find a hyperbolic equation from a design:

1. Assume the center of the hyperbola is at the origin $(0, 0)$ and it opens horizontally, using the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.
2. The narrowest distance between the sides of the tower corresponds to the distance between vertices, $2a$.
3. Use the tower's other dimensions, such as its height and the diameter of its top, to find a point $(x, y)$ that lies on the curve.
4. Substitute the values of $a$, $x$, and $y$ into the standard form equation to solve for $b^2$.

Examples

• A cooling tower is 100 meters tall. The diameter at the top is 50 meters. The sides are 40 meters apart at their closest point. Centered at the origin, $2a=40 \implies a=20$. A point at the top is $(25, 50)$. Substitute into $\frac{x^2}{20^2} - \frac{y^2}{b^2} = 1$ to find $b^2$: $\frac{25^2}{400} - \frac{50^2}{b^2} = 1 \implies 1.5625 - 1 = \frac{2500}{b^2} \implies b^2 = \frac{2500}{0.5625} \approx 4444.44$.
• Two radar stations are 20 miles apart on an east-west line. A plane's signal indicates it is 10 miles closer to one station than the other. The stations are foci, so $2c=20 \implies c=10$. The difference in distance is $2a=10 \implies a=5$. The plane lies on a hyperbola with $b^2 = 10^2 - 5^2 = 75$. The equation is $\frac{x^2}{25} - \frac{y^2}{75} = 1$.
• A sculpture in the shape of a hyperbola is 8 feet tall. Its narrowest width is 2 feet, and its top width is 6 feet. Centered at the origin, $2a=2 \implies a=1$. A point on the top edge is $(3, 4)$. Substituting into $\frac{x^2}{1^2} - \frac{y^2}{b^2} = 1$ gives $9 - \frac{16}{b^2} = 1$, so $b^2 = \frac{16}{8}=2$. The equation is $x^2 - \frac{y^2}{2} = 1$.

Explanation

Hyperbolas are used in engineering because their shape is structurally strong and efficient. By placing a real-world object like a cooling tower on a coordinate plane, its dimensions can be used to create a precise mathematical equation describing its design.