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Introduction to Hyperbolas

The last conic section is the hyperbola. Hyperbolas are formed when a slice is made through both parts of a double cone. The graph of a hyperbola comes in two pieces called branches . Like ellipses, hyperbolas have a center, two foci, and two vertices. Hyperbolas also have two slant asymptotes. The definition of a hyperbola involves the distance between points on the graph and two fixed points.

DEFINITION: A hyperbola is the set of all points such that the difference of the distance between a point and two fixed points (the foci) is constant.

For example, the foci for Conic Sections Hyperbolas are (−5, 0) and (5, 0). For any point on the hyperbola, the distance between this point and one focus minus the distance between the same point and the other focus is 6. Two points on the hyperbola are .

Conic Sections Hyperbolas

And

Conic Sections Hyperbolas

Equations of hyperbolas come in one of two forms.

Conic Sections Hyperbolas

If the x ² term is positive, one branch opens to the left and the other to the right. If the y ² term is positive, one branch opens up and the other down. The formulas for these two forms are in Figures 12.21 and 12.22.

Conic Sections Hyperbolas

Fig. 12.21

Conic Sections Hyperbolas

Fig. 12.22

Sketching Hyperbolas on a Graph

We can sketch a hyperbola by plotting the vertices and sketching the asymptotes, using dashed lines. We should also plot two points to the left and two points to the right of the vertices.

Example

Sketch the graph for .
Because y ² is positive, we will use the information in Figure 12.22 . The center is (0, 0), a = 2, and b = 1. The vertices are ( h , k + a ) = (0, 0 + 2) = (0, 2) and ( h , k − a ) = (0, 0 − 2) = (0, −2). The asymptote formulas are . Using our numbers for h, k, a , and b , we have y = −2 x and y = 2x . We will use x = 4 and x = −4 for our extra points. If we let x = 4 or x = −4, we get two y -values, . These give us four more . (see Figure 12.23.)

Conic Sections Hyperbolas

Fig. 12.22

Conic Sections Hyperbolas

Fig. 12.23

Finding the Center, Vertices, Foci, and Asymptotes of a Hyperbola

In the next problem, we will find the center, vertices, foci, and asymptotes for given hyperbolas. Once we have determined whether x ² is positive or y ² is positive, we can decide on which formulas to use, those in Figure 12.21 or Figure 12.22.

Conic Sections Hyperbolas

Fig. 12.21

Conic Sections Hyperbolas

Fig. 12.22

Examples

Find the center, vertices, foci, and asymptotes for the hyperbola.

Because x ² is positive, we will use the information in Figure 12.21 .

h = −7, k = −4, a = 6, b = 8,

Center: (−7, −4)

Vertices: ( h − a , k ) = (−7 −6, −4) = (−13, −4) and ( h + a , k ) = (−7 + 6, −4) = (−1, −4)

Foci: ( h − c , k ) = (−7 − 10, −4) = (−17, −4) and ( h + c , k ) = (−7 + 10, −4) = (3, −4)

Asymptotes: and

Because y ² is positive, we need to use the information in Figure 12.22 .

h = 1, k = 0, a = 12, b = 5,

Center: (1, 0)

Vertices: ( h , k − a ) = (1, 0−12) = (1, −12) and ( h , k + a ) = (1, 0 + 12) = (1, 12)

Foci: ( h , k − c ) = (1, 0 − 13) = (1, −13) and ( h , k + c ) = (1, 0 + 13) = (1, 13)

Asymptotes: and

Matching Hyperbola Equations with Graphs

In the next problem, we will match equations of hyperbolas with their graphs. Being able to identify the vertices will not be enough. We will also need to use the equations of the asymptotes to find b (we will know a from the vertices). Because the center of each hyperbola will be at (0, 0), the asymptotes will either be Conic Sections Hyperbolas and .

Examples

Match the equation with its graph in Figures 12.24–12.27.

Conic Sections Hyperbolas

Fig. 12.24

Conic Sections Hyperbolas

Fig. 12.25

Conic Sections Hyperbolas

Fig. 12.26

Conic Sections Hyperbolas

Fig. 12.27

The vertices are (−2, 0) and (2, 0). The slopes of the asymptotes are −1 and 1. The graph is in Figure 12.25 .
The vertices are (−2, 0) and (2, 0). The slopes of the asymptotes are and . The graph is in Figure 12.26 .
The vertices are (0, −2) and (0, 2). The slopes of the asymptotes are −1 and 1. The graph is in Figure 12.27 .
The vertices are (0, −2) and (0, 2). The slopes of the asymptotes are −2 and 2. The graph is in Figure 12.24 .

Finding the Equation for a Hyperbola

We can find the equation for a hyperbola when we know some points or a point and the asymptotes. If we have the vertices and foci, then finding an equation for a hyperbola will be similar to finding an equation for an ellipse. If we are given the vertices and asymptotes or foci and asymptotes, we will need to use the slope of one of the asymptotes to find either a or b (we will know one but not the other from the vertices or foci). The first thing we need to decide is which formulas to use—those in Figures 12.21 or Figure 12.22. If the vertices or foci are on the same horizontal line (the y -coordinates are the same), we will use Figure 12.21. If they are on the same vertical line (the x -coordinates are the same), we will use Figure 12.22.

Conic Sections Hyperbolas

Fig. 12.21

Conic Sections Hyperbolas

Fig. 12.22

Examples

Find an equation for the hyperbola.

The vertices are (3, −1) and (3, 7) and is an asymptote. The vertices are on the same vertical line, so we need to use the information in Figure 12.22. The vertices are ( h , k − a ) = (3, −1) and ( h , k + a ) = (3, 7). This gives us h = 3, k − a = −1 and k + a = 7.

Conic Sections Hyperbolas

The center is (3, 3) and a = 4. Once we have b , we will be done. The slope of one of the asymptotes in Figure 12.22 is Conic Sections Hyperbolas , so we have , so b = 3. The equation is

Conic Sections Hyperbolas

The vertices are (−8, 5) and (4, 5), and the foci are (−12, 5) and (8, 5). The vertices and foci are on the same horizontal line, so we need to use the information in Figure 12.21. The vertices are ( h − a , k ) = (−8, 5) and ( h + a , k ) = (4, 5). Now we know k = 5 and we have the system h − a = −8 and h + a = 4.

Conic Sections Hyperbolas

−2 − a = −8 Let h = −2 in h − a = − 8

a = 6

A focus is ( h − c , k ) = (−2 − c , 5) = (−12, 5), which gives us −2 − c = –12. Now that we see that c = 10, we can put this and a = 6 in Conic Sections Hyperbolas to find b .

Conic Sections Hyperbolas

The equation is

Conic Sections Hyperbolas

Hyperbola Practice Problems

Practice

Find the center, vertices, foci, and asymptotes for
Find the center, vertices, foci, and asymptotes for

Conic Sections Hyperbolas

Find an equation for the hyperbola having vertices (−4, 2) and (12, 2) and foci (−6, 2) and (14, 2).
Find an equation for the hyperbola having vertices (−8, 0) and (−4, 0) and with an asymptote .

In Problems 5–7, match the graphs in Figures 12.28–12.30 with their equations.

( y − 1) ² − ( x − 1) ² = 1
( x − 1) ² − ( y − 1) ² = 1

Conic Sections Hyperbolas

Fig. 12.28

Conic Sections Hyperbolas

Fig. 12.29

Conic Sections Hyperbolas

Fig. 12.30

Solutions

h = 5, k = 0, a = 4, b = 3, and

Center: (5, 0)

Vertices: (h, k − a ) = (5, 0 − 4) = (5, −4) and (h, k + a) = (5, 0 + 4) = (5, 4)

Foci: ( h , k − c ) = (5, 0−5) = (5, −5) and ( h , k + c ) = (5, 0 + 5) = (5, 5)

Asymptotes: and
h = −8, k = −6, a = 7, b = 24, and

Center: (−8, −6)

Vertices: ( h − a , k ) = (−8 − 7, −6) = (−15, −6) and ( h + a , k ) = (−8 + 7, −6) = (−1, −6)

Foci: ( h − c , k ) = (−8 − 25, −6) = (−33, −6) and ( h + c , k ) = (−8 + 25, −6) = (17, −6)

Asymptotes: and
The vertices are (−4, 2) and (12, 2), which gives us k = 2 and ( h − a , k ) = (−4, 2) and ( h + a , k ) = (12, 2).

A focus is (−6, 2), which gives us ( h − c , k ) = (−6, 2) and h − c = 4 − c = −6. Solving 4 − c = −6 gives us c = 10. We can find b by letting a = 8 and c = 10 in .

The equation is
( h − a , k ) = (−8, 0) and ( h + a , k ) = (−4, 0), so k = 0 and we have the following system.

The slope of an asymptote is , so . The equation is
Figure 12.30
Figure 12.28
Figure 12.29

Conics on a Graphing Calculator

In order to use a graphing calculator to graph a conic section, the equation probably needs to be entered as two separate functions. For example, the graph of y ² = x could be entered as Conic Sections Hyperbolas . To use a graphing calculator to graph a conic section that is not a function, solve for y . When taking the square root of both sides, we use a “±” symbol on one of the sides. It is this sign that gives us two separate equations.

Examples

Solve for y .

Conic Sections Hyperbolas

Conic Equations - General and Standard Forms

Equations of conic sections do not always come in the convenient forms we have been using. Sometimes they come in the general form Ax ² + Bxy + Cy ² + Dx + Ey + F = 0. When A and C are equal (and B = 0), the graph is a circle. If A and C are positive and not equal (and B = 0), the graph is an ellipse. If A and C have different signs (and B = 0), the graph is a hyperbola. If only one of A or C is nonzero (and B = 0), the graph is a parabola. There are some conic sections whose entire graph is one point. These are called degenerate conics . We can rewrite an equation in the general form in the standard form (the form we have been using) by completing the square.

Examples

Rewrite the equation in standard form.

x ² − 2 x − 4 y = 11
Because there is no y ² term, the graph of this equation is a parabola that opens up or down. The standard equation is ( x − h ) ² = 4 p ( y − k ). We need to have the x terms on one side of the equation and the other terms on the other side.

Conic Sections Hyperbolas

−9 x ² + 16 y ² − 18 x − 160 y + 247 = 0
Because the signs on x ² and y ² are different, the graph of this equation is a hyperbola. The standard form for this equation is .

Conic Sections Hyperbolas

Conic Equations Practice Problems

Practice

Solve for y
Solve for y
Rewrite the equation in standard form: 36 x ² + 9 y ² − 216 x − 72 y + 144 = 0.

Solutions

Conic Sections Hyperbolas

Practice problems for this concept can be found at: Conic Sections Practice Test.

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Hyperbolas Help (page 2)

Introduction to Hyperbolas

Sketching Hyperbolas on a Graph

Example

Finding the Center, Vertices, Foci, and Asymptotes of a Hyperbola

Examples

Matching Hyperbola Equations with Graphs

Examples

Finding the Equation for a Hyperbola

Examples

Hyperbola Practice Problems

Practice

Solutions

Conics on a Graphing Calculator

Examples

Conic Equations - General and Standard Forms

Examples

Conic Equations Practice Problems

Practice

Solutions

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