LIAL HORNSBY SCHNEIDER COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER

Hyperbolas 6.3 Equations and Graphs of Hyperbolas Translated Hyperbolas Eccentricity

Equations and Graphs of Hyperbolas An ellipse was defined as the set of all points in a plane the sum of whose distances from two fixed points is a constant. A hyperbola is defined similarly.

Hyperbola A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances from two fixed points is constant. The two fixed points are called the foci of the hyperbola.

Equations and Graphs of Hyperbolas Suppose a hyperbola has center at the origin and foci at F′(– c, 0) and F(c, 0). The midpoint of the segment F′F is the center of the hyperbola and the points V′(– a, 0) and V(a, 0) are the vertices of the hyperbola. The line segment V′V is the transverse axis of the hyperbola.

Equations and Graphs of Hyperbolas As with the ellipse, so the constant in the definition is 2a, and for any point P(x, y) on the hyperbola.

Equations and Graphs of Hyperbolas The distance formula and algebraic manipulation similar to that used for finding an equation for an ellipse produce the result

Equations and Graphs of Hyperbolas Letting b2 = c2 – a2 gives as an equation of the hyperbola. Letting y = 0 shows that the x-intercepts are a.

Equations and Graphs of Hyperbolas If x = 0, the equation becomes Let x = 0. Multiply by – b2. which has no real number solutions, showing that this hyperbola has no y-intercepts.

Equations and Graphs of Hyperbolas Starting with the equation for a hyperbola and solving for y, we get Subtract 1; add

Equations and Graphs of Hyperbolas Write the left side as a single fraction. Remember both the positive and negative square roots. Take square roots on both sides; multiply by b.

Equations and Graphs of Hyperbolas If x2 is very large in comparison to a2, the difference x2 – a2 would be very close to x2. If this happens, then the points satisfying the final equation above would be very close to one of the lines

Equations and Graphs of Hyperbolas Thus, as x gets larger and larger, the points of the hyperbola get closer and closer to the lines

Equations and Graphs of Hyperbolas These lines, called asymptotes of the hyperbola, are useful when sketching the graph.

USING ASYMPTOTES TO GRAPH A HYPERBOLA Example 1 Graph Sketch the asymptotes, and find the coordinates of the vertices and foci. Give the domain and range. Algebraic Solution For this hyperbola, a = 5 and b = 7.

y = 7. Choosing x = – 5 also gives y = 7. USING ASYMPTOTES TO GRAPH A HYPERBOLA Example 1 With these values Asymptotes If we choose x = 5, then y = 7. Choosing x = – 5 also gives y = 7. These four points — (5, 7), (5,– 7), (– 5, 7) and (– 5, – 7) — are the corners of the rectangle shown here.

USING ASYMPTOTES TO GRAPH A HYPERBOLA Example 1 The extended diagonals of this rectangle, called the fundamental rectangle, are the asymptotes of the hyperbola. Since a = 5, the vertices of the hyperbola are (5, 0) and (– 5, 0).

We find the foci by letting USING ASYMPTOTES TO GRAPH A HYPERBOLA Example 1 We find the foci by letting so Therefore, the foci are and The domain is (– , – 5]  [5, ), and the range is (– , ).

Note When graphing hyperbolas, remember that the fundamental rectangle and the asymptotes are not actually parts of the graph. They are simply aids in sketching the graph.

USING ASYMPTOTES TO GRAPH A HYPERBOLA While a > b for an ellipse, examples would show that for hyperbolas, it is possible that a > b, a < b, or a = b. If the foci of a hyperbola are on the y-axis, the equation of the hyperbola has the form with asymptotes

Caution If the foci of a hyperbola are on the x-axis, the asymptotes have equations while foci on the y-axis lead to asymptotes To avoid errors, write the equation of the hyperbola in either the form and replace 1 with 0. Solving the resulting equation for y produces the proper equations for the asymptotes. or

The hyperbola with center at the origin and equation STANDARD FORMS OF EQUATIONS FOR HYPERBOLAS The hyperbola with center at the origin and equation has vertices (a, 0), asymptotes and foci (c, 0), where c2 = a2 +b 2.

The hyperbola with center at the origin and equation STANDARD FORMS OF EQUATIONS FOR HYPERBOLAS The hyperbola with center at the origin and equation has vertices (0, a), asymptotes and foci (0, c), where c2 = a2 +b 2.

Give the domain and range. GRAPHING A HYPERBOLA Example 2 Graph Give the domain and range. Solution Divide by 100; standard form. This hyperbola is centered at the origin, has foci on the y-axis, and has vertices (0, 2) and (0, – 2). To find the equations of the asymptotes, replace 1 with 0.

Remember both the positive and negative square roots.. GRAPHING A HYPERBOLA Example 2 Add Remember both the positive and negative square roots.. Multiply by 4. Square root property.

GRAPHING A HYPERBOLA Example 2 To graph the asymptotes, use the points (5, 2), (5,– 2), (– 5, 2) and (– 5, – 2) to determine the fundamental rectangle. The diagonals of this rectangle are the asymptotes for the graph, as shown here. The domain of the relation is (– , ), and the range is (–  ,– 2]  [2,  ).

Translated Hyperbolas In the graph of each hyperbola considered so far, the center is the origin and the asymptotes pass through the origin. This feature holds in general; the asymptotes of any hyperbola pass through the center of the hyperbola. Like an ellipse, a hyperbola can have its center translated away from the origin.

Solution Example 3 Graph Give the domain and range. GRAPHING A HYPERBOLA TRANSLATED AWAY FROM THE ORIGIN Example 3 Graph Give the domain and range. Solution This equation represents a hyperbola centered at (– 3, – 2). For this vertical hyperbola, a = 3 and b = 2. The x-values of the vertices are – 3. Locate the y-values of the vertices by taking the y-value of the center, – 2 , and adding and subtracting 3. Thus, the vertices are (– 3, 1) and (– 3, – 5). The asymptotes have slopes and pass through the center (– 3,– 2).

Solution Example 3 Graph Give the domain and range. GRAPHING A HYPERBOLA TRANSLATED AWAY FROM THE ORIGIN Example 3 Graph Give the domain and range. Solution The equations of the asymptotes can be found either by using the point-slope form of the equation of a line or by replacing 1 with 0 in the equation of the hyperbola as was done in Example 2.

Solution Example 3 Graph Give the domain and range. GRAPHING A HYPERBOLA TRANSLATED AWAY FROM THE ORIGIN Example 3 Graph Give the domain and range. Solution Using the point-slope form, we get Point-slope form. Solve for y.

Solution Example 3 Graph Give the domain and range. GRAPHING A HYPERBOLA TRANSLATED AWAY FROM THE ORIGIN Example 3 Graph Give the domain and range. Solution The graph is shown here. The domain of the relation is (– , ), and the range is (–  ,– 5]  [1,  ).

Eccentricity If we apply the definition of eccentricity from the previous section to the hyperbola, we get Since c > a, we have e > 1. Narrow hyperbolas have e near 1, and wide hyperbolas have large e. See the figures on the next slide.

Eccentricity

Solution Example 4 Find the eccentricity of the hyperbola FINDING ECCENTRICITY FROM THE EQUATION OF A HYPERBOLA Example 4 Find the eccentricity of the hyperbola Solution Here a2 = 9; thus, a = 3, and

Solution FINDING THE EQUATION OF A HYPERBOLA Example 5 Find the equation of the hyperbola with eccentricity 2 and foci at (– 9, 5) and (– 3, 5). Solution Since the foci have the same y-coordinate, the line through them, and therefore the hyperbola, is horizontal. The center of the hyperbola is halfway between the two foci at (– 6, 5). The distance from each focus to the center is c = 3.

Solution FINDING THE EQUATION OF A HYPERBOLA Example 5 Find the equation of the hyperbola with eccentricity 2 and foci at (– 9, 5) and (– 3, 5). Solution Since Thus,

Solution FINDING THE EQUATION OF A HYPERBOLA Example 5 Find the equation of the hyperbola with eccentricity 2 and foci at (– 9, 5) and (– 3, 5). Solution The equation of the hyperbola is or Simplify complex fractions.

Summary Conic Eccentricity Parabola e = 1 Circle e = 0 Ellipse Hyperbola and and