The Hyperbola

What do paths of comets, supersonic booms, ancient Grecian pillars, and natural draft cooling towers have in common? They can all be modeled by the same type of conic. For instance, when something moves faster than the speed of sound, a shock wave in the form of a cone is created. A portion of a conic is formed when the wave intersects the ground, resulting in a sonic boom.

A shock wave intersecting the ground forms a portion of a conic and results in a sonic boom.

Most people are familiar with the sonic boom created by supersonic aircraft, but humans were breaking the sound barrier long before the first supersonic flight. The crack of a whip occurs because the tip is exceeding the speed of sound. The bullets shot from many firearms also break the sound barrier, although the bang of the gun usually supersedes the sound of the sonic boom.

Locating the Vertices and Foci of a Hyperbola

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. This intersection produces two separate unbounded curves that are mirror images of each other.

A hyperbola.

Like the ellipse, the hyperbola can also be defined as a set of points in the coordinate plane. A hyperbola is 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.

Notice that the definition of a hyperbola is very similar to that of an ellipse. The distinction is that the hyperbola is defined in terms of the difference of two distances, whereas the ellipse is defined in terms of the sum of two distances.

As with the ellipse, every hyperbola has two axes of symmetry. The transverse axis is a line segment that passes through the center of the hyperbola and has vertices as its endpoints. The foci lie on the line that contains the transverse axis. The conjugate axis is perpendicular to the transverse axis and has the co-vertices as its endpoints. The center of a hyperbola is the midpoint of both the transverse and conjugate axes, where they intersect. Every hyperbola also has two asymptotes that pass through its center. As a hyperbola recedes from the center, its branches approach these asymptotes. The central rectangle of the hyperbola is centered at the origin with sides that pass through each vertex and co-vertex; it is a useful tool for graphing the hyperbola and its asymptotes. To sketch the asymptotes of the hyperbola, simply sketch and extend the diagonals of the central rectangle.

Key features of the hyperbola.

In this section, we will limit our discussion to hyperbolas that are positioned vertically or horizontally in the coordinate plane; the axes will either lie on or be parallel to the $x$- and $y$-axes. We will consider two cases: those that are centered at the origin, and those that are centered at a point other than the origin.

Deriving the Equation of a Hyperbola Centered at the Origin

Let $(-c,0)$ and $(c,0)$ be the foci of a hyperbola centered at the origin. The hyperbola is the set of all points $(x,y)$ such that the difference of the distances from $(x,y)$ to the foci is constant.

If $(a,0)$ is a vertex of the hyperbola, the distance from $(-c,0)$ to $(a,0)$ is $a-(-c)=a+c$. The distance from $(c,0)$ to $(a,0)$ is $c-a$. The difference of the distances from the foci to the vertex is

If $(x,y)$ is a point on the hyperbola, we can define the following variables:

By definition of a hyperbola, $d_2-d_1$ is constant for any point $(x,y)$ on the hyperbola. We know that the difference of these distances is $2a$ for the vertex $(a,0)$. It follows that $d_2-d_1=2a$ for any point on the hyperbola. As with the derivation of the equation of an ellipse, we will begin by applying the distance formula. The rest of the derivation is algebraic. Compare this derivation with the one from the previous section for ellipses.

This equation defines a hyperbola centered at the origin with vertices $(\pm a,0)$ and co-vertices $(0,\pm b)$.

(a) Horizontal hyperbola with center $(0,0)$.

(b) Vertical hyperbola with center $(0,0)$.

Example. Identify the vertices and foci of the hyperbola with equation $\tfrac{y^2}{49}-\tfrac{x^2}{32}=1$.

Solution. The equation has the form $\tfrac{y^2}{a^2}-\tfrac{x^2}{b^2}=1$, so the transverse axis lies on the $y$-axis. The hyperbola is centered at the origin, so the vertices serve as the $y$-intercepts of the graph. To find the vertices, set $x=0$, and solve for $y$.

The foci are located at $(0,\pm c)$. Solving for $c$,

Therefore, the vertices are located at $(0,\pm7)$, and the foci are located at $(0,\pm9)$.

Writing Equations of Hyperbolas in Standard Form

Just as with ellipses, writing the equation for a hyperbola in standard form allows us to calculate the key features: its center, vertices, co-vertices, foci, asymptotes, and the lengths and positions of the transverse and conjugate axes. Conversely, an equation for a hyperbola can be found given its key features. We begin by finding standard equations for hyperbolas centered at the origin. Then we will turn our attention to finding standard equations for hyperbolas centered at some point other than the origin.

Hyperbolas Centered at the Origin

Reviewing the standard forms given for hyperbolas centered at $(0,0)$, we see that the vertices, co-vertices, and foci are related by the equation $c^2=a^2+b^2$. Note that this equation can also be rewritten as $b^2=c^2-a^2$. This relationship is used to write the equation for a hyperbola when given the coordinates of its foci and vertices.

Example. What is the standard form equation of the hyperbola that has vertices $(\pm6,0)$ and foci $(\pm2\sqrt{10},0)$?

Solution. The vertices and foci are on the $x$-axis. Thus, the equation for the hyperbola will have the form $\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}=1$.

The vertices are $(\pm6,0)$, so $a=6$ and $a^2=36$.

The foci are $(\pm2\sqrt{10},0)$, so $c=2\sqrt{10}$ and $c^2=40$.

Solving for $b^2$, we have

Finally, we substitute $a^2=36$ and $b^2=4$ into the standard form of the equation, $\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}=1$. The equation of the hyperbola is $\tfrac{x^2}{36}-\tfrac{y^2}{4}=1$, as shown below.

Hyperbolas Not Centered at the Origin

Like the graphs for other equations, the graph of a hyperbola can be translated. If a hyperbola is translated $h$ units horizontally and $k$ units vertically, the center of the hyperbola will be $(h,k)$. This translation results in the standard form of the equation we saw previously, with $x$ replaced by $(x-h)$ and $y$ replaced by $(y-k)$.

(a) Horizontal hyperbola with center $(h,k)$.

(b) Vertical hyperbola with center $(h,k)$.

Like hyperbolas centered at the origin, hyperbolas centered at a point $(h,k)$ have vertices, co-vertices, and foci that are related by the equation $c^2=a^2+b^2$. We can use this relationship along with the midpoint and distance formulas to find the standard equation of a hyperbola when the vertices and foci are given.

Example. What is the standard form equation of the hyperbola that has vertices at $(0,-2)$ and $(6,-2)$ and foci at $(-2,-2)$ and $(8,-2)$?

Solution. The $y$-coordinates of the vertices and foci are the same, so the transverse axis is parallel to the $x$-axis. Thus, the equation of the hyperbola will have the form

First, we identify the center, $(h,k)$. The center is halfway between the vertices $(0,-2)$ and $(6,-2)$. Applying the midpoint formula, we have

Next, we find $a^2$. The length of the transverse axis, $2a$, is bounded by the vertices. So, we can find $a^2$ by finding the distance between the $x$-coordinates of the vertices.

Now we need to find $c^2$. The coordinates of the foci are $(h\pm c,k)$. So $(h-c,k)=(-2,-2)$ and $(h+c,k)=(8,-2)$. We can use the $x$-coordinate from either of these points to solve for $c$. Using the point $(8,-2)$, and substituting $h=3$,

Next, solve for $b^2$ using the equation $b^2=c^2-a^2$:

Finally, substitute the values found for $h,k,a^2,$ and $b^2$ into the standard form of the equation.

Graphing Hyperbolas Centered at the Origin

When we have an equation in standard form for a hyperbola centered at the origin, we can interpret its parts to identify the key features of its graph: the center, vertices, co-vertices, asymptotes, foci, and lengths and positions of the transverse and conjugate axes. To graph hyperbolas centered at the origin, we use the standard form $\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}=1$ for horizontal hyperbolas and the standard form $\tfrac{y^2}{a^2}-\tfrac{x^2}{b^2}=1$ for vertical hyperbolas.

Example. Graph the hyperbola given by the equation $\tfrac{y^2}{64}-\tfrac{x^2}{36}=1$. Identify and label the vertices, co-vertices, foci, and asymptotes.

Solution. The standard form that applies to the given equation is $\tfrac{y^2}{a^2}-\tfrac{x^2}{b^2}=1$. Thus, the transverse axis is on the $y$-axis.

The coordinates of the vertices are $(0,\pm a)=(0,\pm\sqrt{64})=(0,\pm8)$

The coordinates of the co-vertices are $(\pm b,0)=(\pm\sqrt{36},0)=(\pm6,0)$

The coordinates of the foci are $(0,\pm c)$, where $c=\pm\sqrt{a^2+b^2}$. Solving for $c$, we have

Therefore, the coordinates of the foci are $(0,\pm10)$

The equations of the asymptotes are $y=\pm\tfrac{a}{b}x=\pm\tfrac{8}{6}x=\pm\tfrac{4}{3}x$

Plot and label the vertices and co-vertices, and then sketch the central rectangle. Sides of the rectangle are parallel to the axes and pass through the vertices and co-vertices. Sketch and extend the diagonals of the central rectangle to show the asymptotes. The central rectangle and asymptotes provide the framework needed to sketch an accurate graph of the hyperbola. Label the foci and asymptotes, and draw a smooth curve to form the hyperbola, as shown below.

Graphing Hyperbolas Not Centered at the Origin

Graphing hyperbolas centered at a point $(h,k)$ other than the origin is similar to graphing ellipses centered at a point other than the origin. We use the standard forms $\tfrac{(x-h)^2}{a^2}-\tfrac{(y-k)^2}{b^2}=1$ for horizontal hyperbolas, and $\tfrac{(y-k)^2}{a^2}-\tfrac{(x-h)^2}{b^2}=1$ for vertical hyperbolas. From these standard form equations we can easily calculate and plot key features of the graph: the coordinates of its center, vertices, co-vertices, and foci; the equations of its asymptotes; and the positions of the transverse and conjugate axes.

Example. Graph the hyperbola given by the equation $9x^2-4y^2-36x-40y-388=0$. Identify and label the center, vertices, co-vertices, foci, and asymptotes.

Solution. Start by expressing the equation in standard form. Group terms that contain the same variable, and move the constant to the opposite side of the equation.

Factor the leading coefficient of each expression.

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

Rewrite as perfect squares.

Divide both sides by the constant term to place the equation in standard form.

The standard form that applies to the given equation is $\tfrac{(x-h)^2}{a^2}-\tfrac{(y-k)^2}{b^2}=1$, where $a^2=36$ and $b^2=81$, or $a=6$ and $b=9$. Thus, the transverse axis is parallel to the $x$-axis. It follows that:

• the center of the hyperbola is $(h,k)=(2,-5)$
• the coordinates of the vertices are $(h\pm a,k)=(2\pm6,-5)$, or $(-4,-5)$ and $(8,-5)$
• the coordinates of the co-vertices are $(h,k\pm b)=(2,-5\pm9)$, or $(2,-14)$ and $(2,4)$
• the coordinates of the foci are $(h\pm c,k)$, where $c=\pm\sqrt{a^2+b^2}$. Solving for $c$, we have

Therefore, the coordinates of the foci are $(2-3\sqrt{13},-5)$ and $(2+3\sqrt{13},-5)$.

The equations of the asymptotes are $y=\pm\tfrac{b}{a}(x-h)+k=\pm\tfrac{3}{2}(x-2)-5$.

Next, we plot and label the center, vertices, co-vertices, foci, and asymptotes and draw smooth curves to form the hyperbola, as shown below.

Solving Applied Problems Involving Hyperbolas

As we discussed at the beginning of this section, hyperbolas have real-world applications in many fields, such as astronomy, physics, engineering, and architecture. The design efficiency of hyperbolic cooling towers is particularly interesting. Cooling towers are used to transfer waste heat to the atmosphere and are often touted for their ability to generate power efficiently. Because of their hyperbolic form, these structures are able to withstand extreme winds while requiring less material than any other forms of their size and strength. For example, a 500-foot tower can be made of a reinforced concrete shell only 6 or 8 inches wide!

The first hyperbolic towers were designed in 1914 and were 35 meters high. Today, the tallest cooling towers are in France, standing a remarkable 170 meters tall. In the following example we will use the design layout of a cooling tower to find a hyperbolic equation that models its sides.

Example. The design layout of a cooling tower is shown below. The tower stands $179.6$ meters tall. The diameter of the top is $72$ meters. At their closest, the sides of the tower are $60$ meters apart.

Project design for a natural draft cooling tower.

Find the equation of the hyperbola that models the sides of the cooling tower. Assume that the center of the hyperbola—indicated by the intersection of dashed perpendicular lines in the figure—is the origin of the coordinate plane. Round final values to four decimal places.

Solution. We are assuming the center of the tower is at the origin, so we can use the standard form of a horizontal hyperbola centered at the origin: $\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}=1$, where the branches of the hyperbola form the sides of the cooling tower. We must find the values of $a^2$ and $b^2$ to complete the model.

First, we find $a^2$. Recall that the length of the transverse axis of a hyperbola is $2a$. This length is represented by the distance where the sides are closest, which is given as $60$ meters. So, $2a=60$. Therefore, $a=30$ and $a^2=900$.

To solve for $b^2$, we need to substitute for $x$ and $y$ in our equation using a known point. To do this, we can use the dimensions of the tower to find some point $(x,y)$ that lies on the hyperbola. We will use the top right corner of the tower to represent that point. Since the $y$-axis bisects the tower, our $x$-value can be represented by the radius of the top, or $36$ meters. The $y$-value is represented by the distance from the origin to the top, which is given as $79.6$ meters. Therefore,

The sides of the tower can be modeled by the hyperbolic equation

Project design for a natural draft cooling tower.

Key equations

Key concepts

• A hyperbola is 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 standard form of a hyperbola can be used to locate its vertices and foci.
• When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form.
• When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola.
• Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the dimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides.

Practice

Locate a hyperbola’s vertices and foci

Write equations of hyperbolas in standard form

Graph hyperbolas centered at the origin

Graph hyperbolas not centered at the origin

Solve applied problems involving hyperbolas

This section is adapted from Precalculus 2e, Section 10.2: The Hyperbola by Jay Abramson and OpenStax, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: omitted a coreq-skills block the pinned CNXML prepends before the section proper (its own review of the Distance Formula and graphing a hyperbola centered at the origin, keyed to Intermediate Algebra 11.1.1/11.4.1, complete with its own worked example and two “Practice Makes Perfect” exercises) — confirmed against the rendered PDF, page 1026 (true PDF index 1036), where the section heading “10.2 The Hyperbola” runs directly into the sonic-boom introduction with no corequisite-skills material between them; the section’s own five learning objectives, from the module’s abstract, are unaffected. Omitted the credit photograph of the Drax power station cooling towers (Figure 10, credit: Les Haines, Flickr), keeping the sentences that surround it. Recreated every instructional figure as an accessible spec-first SVG: the sonic-boom schematic (Figure 1, a jet trailing nested wavefront lines whose ground intersection traces a hyperbola) and the double-cone conic-section diagram (Figure 2) as kind="figure" figures, the latter built with the parent’s cone-schematic generator (hyperbola panel); the generic key-features diagram (Figure 3) and the foci-distance diagram ($d_1$, $d_2$, Figure 4) as labeled graphs built from the hyperbolas primitive with dashed lines/segments for the asymptotes and central rectangle; the origin- and $(h,k)$-centered standard-form summary pairs (Figures 5 and 7) as two side-by-side apfigures each, one per orientation; and every worked example’s answer graph (Figures 6, 8, 9) the same way, with $c^2=a^2+b^2$ verified by script for every labelled focus and every asymptote line confirmed to pass through its central rectangle’s corner; and the two cooling-tower design diagrams (Figures 11 and 12) as kind="figure" profiles whose two sides are sampled exactly from the tower’s own hyperbola ($x=\pm a\sqrt{1+y^2/b^2}$ between the base and the top), with the waist dashed and the printed dimensions on arrowed dimension lines. Confirmed source defect (reported for errata, not corrected here since it affects no graded content): the CNXML alt text for Figure 5 (Figure_10_02_005, the $(h,k)$ standard-forms figure) describes a horizontal parabola with a directrix and latus rectum — the alt text of a parabola figure from a different section — while the figure’s own caption and the surrounding prose both describe, and the rendered PDF page confirms, the two-panel horizontal/vertical hyperbola-with-center-$(h,k)$ pair reproduced above; the recreation follows the caption, prose, and PDF, not the mismatched alt text. Confirmed source defect, silently corrected in prose (harmless, ungraded): the CNXML and PDF both read “the center of the ellipse is $(h,k)=(2,-5)$” in Example 5’s key-feature list — a copy-paste leftover from the parallel ellipse-chapter example, since the whole example is otherwise consistently about a hyperbola; rendered here as “the center of the hyperbola” for readability. Confirmed source defect, not corrected (alt text only, no rendered effect): Figure 8’s (Figure_10_02_008) alt text gives Example 5’s second vertex as “$(8,5)$”, dropping the negative sign the algebra and the printed PDF page both carry ($(h+a,k)=(2+6,-5)=(8,-5)$); the figure recreated above and its ariaLabel use the correct $(8,-5)$. Confirmed source defect, worked around: Try It 6’s figure (Figure_10_02_012) alt text states the cooling-tower design’s height-to-waist distance as “79.6 meters from the top” and both the top and waist diameters as “60 meters,” reusing Example 6’s own numbers; the rendered PDF page (1039, true PDF index 1049) shows the actual figure labels $167.082\text{ m}$ total height, $60\text{ m}$ top diameter, $40\text{ m}$ waist diameter, and $67.082\text{ m}$ from the waist to the top — the values used here, matching the source’s own printed answer $\tfrac{x^2}{400}-\tfrac{y^2}{3{,}600}=1$; because that answer’s $b^2=3{,}600$ follows only from the exact height $30\sqrt5$ (the printed $67.082$ m, carried through, gives $b^2\approx3{,}599.9958$), the local question states the height as $30\sqrt5\approx67.082$ m and asks for the exact value. Computing $b^2$ directly from the printed $67.082\text{ m}$ intermediate gives $b^2\approx3{,}599.9958$ rather than the source’s clean $3{,}600$ (the design’s underlying exact height above the waist is $\sqrt{4{,}500}\approx67.08204$, of which $67.082$ is itself a rounded display value) — a sub-$0.01\%$ rounding-path difference; the source’s own $3{,}600$ is shipped as printed. Kept the “Media” callout’s introductory sentence but omitted its four external links, matching house precedent elsewhere in this book. Every retained Try It’s vertices/foci/asymptotes/center ask was split into one fill-in per named quantity — the source’s own Try Its 4 and 5 request four and five quantities respectively, each Try It here keeps three of them (the 2–3 consecutive-question convention for in-page practice), independently re-derived and verified against the printed key. Ten retained Section Exercises (two per objective) were expanded into nineteen interactive Practice components, one group per objective, every one independently re-derived (including by running the arithmetic in Node and replaying every standard-form answer’s printed general-form equation through the real grader under conic-standard-form) rather than read off the source key.