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Five-Minute Check (over Lesson 7-2) Then/Now New Vocabulary Key Concept: Standard Forms of Equations for Hyperbolas Example 1: Graph Hyperbolas in Standard Form Example 2: Graph a Hyperbola Example 3: Write Equations Given Characteristics Example 4: Find the Eccentricity of a Hyperbola Key Concept: Classify Conics Using the Discriminant Example 5: Identify Conic Sections Example 6: Real-World Example: Apply Hyperbolas Lesson Menu

Graph the ellipse given by 4x 2 + y 2 + 16x – 6y – 39 = 0. A. B. C. D. 5-Minute Check 1

Write an equation in standard form for the ellipse with vertices (–3, –1) and (7, –1) and foci (–2, –1) and (6, –1). A. B. C. D. 5-Minute Check 2

Determine the eccentricity of the ellipse given by A. 0.632 B. 0.775 C. 0.845 D. 1.290 5-Minute Check 3

Write an equation in standard form for a circle with center at (–2, 5) and radius 3. A. (x + 2)2 + (y – 5)2 = 3 B. (x + 2)2 + (y – 5)2 = 9 C. (x – 2)2 + (y + 5)2 = 9 D. (x – 2)2 + (y + 5)2 = 3 5-Minute Check 4

Identify the conic section represented by 8x 2 + 5y 2 – x + 6y = 0. A. circle B. ellipse C. parabola D. none of the above 5-Minute Check 5

You analyzed and graphed ellipses and circles. (Lesson 7-2) Analyze and graph equations of hyperbolas. Use equations to identify types of conic sections. Then/Now

hyperbola transverse axis conjugate axis Vocabulary

Key Concept 1

A. Graph the hyperbola given by Graph Hyperbolas in Standard Form A. Graph the hyperbola given by The equation is in standard form with h = 0, k = 0, Example 1

Graph Hyperbolas in Standard Form Graph the center, vertices, foci, and asymptotes. Then make a table of values to sketch the hyperbola. Answer: Example 1

B. Graph the hyperbola given by Graph Hyperbolas in Standard Form B. Graph the hyperbola given by The equation is in standard form with h = 2 and k = –4. Because a2 = 4 and b2 = 9, a = 2 and b = 3. Use the values of a and b to find c. c2 = a2 + b2 Equation relating a, b, and c for a hyperbola c2 = 4 + 9 a2 = 4 and b2 = 9 Solve for c. Example 1

Graph Hyperbolas in Standard Form Use h, k, a, b, and c to determine the characteristics of the hyperbola. Example 1

Graph Hyperbolas in Standard Form Graph the center, vertices, foci, and asymptotes. Then make a table of values to sketch the hyperbola. Answer: Example 1

Graph the hyperbola given by A. B. C. D. Example 1

First, write the equation in standard form. Graph a Hyperbola Graph the hyperbola given by 4x2 – y2 + 24x + 4y = 28. First, write the equation in standard form. 4x2 – y2 + 24x + 4y = 28 Original equation 4x2 + 24x – y2 + 4y = 28 Isolate and group like terms. 4(x2 + 6x) – (y2 – 4y) = 28 Factor. 4(x2 + 6x + 9) – (y2 – 4y + 4) = 28 + 4(9) – 4 Complete the squares. 4(x + 3)2 – (y – 2)2 = 60 Factor and simplify. Example 2

The equation is now in standard form with h = –3, Graph a Hyperbola Divide each side by 60. The equation is now in standard form with h = –3, Example 2

Graph a Hyperbola Graph the center, vertices, foci, and asymptotes. Then make a table of values to sketch the hyperbola. Answer: Example 2

Graph the hyperbola given by 3x2 – y2 – 30x – 4y = –119. A. B. C. D. Example 2

center: (1, –2) Midpoint of segment between foci Write Equations Given Characteristics A. Write an equation for the hyperbola with foci (1, –5) and (1, 1) and transverse axis length of 4 units. Because the x-coordinates of the foci are the same, the transverse axis is vertical. Find the center and the values of a, b, and c. center: (1, –2) Midpoint of segment between foci a = 2 Transverse axis = 2a c = 3 Distance from each focus to center c2 = a2 + b2 Example 3

Write Equations Given Characteristics Answer: Example 3

center: (–3, 4) Midpoint of segment between vertices Write Equations Given Characteristics B. Write an equation for the hyperbola with vertices (–3, 10) and (–3, –2) and conjugate axis length of 6 units. Because the x-coordinates of the foci are the same, the transverse axis is vertical. Find the center and the values of a, b, and c. center: (–3, 4) Midpoint of segment between vertices b = 3 Conjugate axis = 2b a = 6 Distance from each vertex to center Example 3

Write Equations Given Characteristics Answer: Example 3

Write an equation for the hyperbola with foci at (13, –3) and (–5, –3) and conjugate axis length of 12 units. A. B. C. D. Example 3

Find c and then determine the eccentricity. Find the Eccentricity of a Hyperbola Find c and then determine the eccentricity. c2 = a2 + b2 Equation relating a, b, and c c2 = 32 + 25 a2 = 32 and b2 = 25 Simplify. Example 4

Eccentricity equation Find the Eccentricity of a Hyperbola Eccentricity equation Simplify. The eccentricity of the hyperbola is about 1.33. Answer: 1.33 Example 4

A. 0.59 B. 0.93 C. 1.24 D. 1.69 Example 4

Key Concept 3

The discriminant is greater than 0, so the conic is a hyperbola. Identify Conic Sections A. Use the discriminant to identify the conic section in the equation 2x2 + y2 – 2x + 5xy + 12 = 0. A is 2, B is 5, and C is 1. Find the discriminant. B2 – 4AC = 52 – 4(2)(1) or 17 The discriminant is greater than 0, so the conic is a hyperbola. Answer: hyperbola Example 5

Identify Conic Sections B. Use the discriminant to identify the conic section in the equation 4x2 + 4y2 – 4x + 8 = 0. A is 4, B is 0, and C is 4. Find the discriminant. B2 – 4AC = 02 – 4(4)(4) or –64 The discriminant is less than 0, so the conic must be either a circle or an ellipse. Because A = C, the conic is a circle. Answer: circle Example 5

The discriminant is 0, so the conic is a parabola. Identify Conic Sections C. Use the discriminant to identify the conic section in the equation 2x2 + 2y2 – 6y + 4xy – 10 = 0. A is 2, B is 4, and C is 2. Find the discriminant. B2 – 4AC = 42 – 4(2)(2) or 0 The discriminant is 0, so the conic is a parabola. Answer: parabola Example 5

Use the discriminant to identify the conic section given by 15 + 6y + y2 = –14x – 3x2. A. ellipse B. circle C. hyperbola D. parabola Example 5

Apply Hyperbolas A. NAVIGATION LORAN (LOng RAnge Navigation) is a navigation system for ships relying on radio pulses that is not dependent on visibility conditions. Suppose LORAN stations E and F are located 350 miles apart along a straight shore with E due west of F. When a ship approaches the shore, it receives radio pulses from the stations and is able to determine that it is 80 miles farther from station F than it is from station E. Find the equation for the hyperbola on which the ship is located. Example 6A

Apply Hyperbolas First, place the two sensors on a coordinate grid so that the origin is the midpoint of the segment between station E and station F. The ship is closer to station E, so it should be in the 2nd quadrant. The two stations are located at the foci of the hyperbola, so c is 175. The absolute value of the difference of the distances from any point on a hyperbola to the foci is 2a. Because the ship is 80 miles farther from station F than station E, 2a = 80 and a = 40. Example 6A

Use the values of a and c to find b2. Apply Hyperbolas Use the values of a and c to find b2. c2 = a2 + b2 Equation relating a, b, and c 1752 = 402 + b2 c = 175 and a = 40 29,025 = b2 Simplify. Example 6A

Apply Hyperbolas Example 6A

Apply Hyperbolas Answer: Example 6A

Apply Hyperbolas B. NAVIGATION LORAN (LOng RAnge Navigation) is a navigation system for ships relying on radio pulses that is not dependent on visibility conditions. Suppose LORAN stations E and F are located 350 miles apart along a straight shore with E due west of F. When a ship approaches the shore, it receives radio pulses from the stations and is able to determine that it is 80 miles farther from station F than it is from station E. Find the exact coordinates of the ship if it is 125 miles from the shore. Example 6B

Apply Hyperbolas Because the ship is 125 miles from the shore, y = 125. Substitute the value of y into the equation and solve for x. Original equation y = 125 Solve. Example 6B

Apply Hyperbolas Since the ship is closer to station E, it is located on the left branch of the hyperbola, and the value of x is about –49.6. Therefore, the coordinates of the ship are (–49.6, 125). Answer: (–49.6, 125) Example 6B

NAVIGATION Suppose LORAN stations S and T are located 240 miles apart along a straight shore with S due north of T. When a ship approaches the shore, it receives radio pulses from the stations and is able to determine that it is 60 miles farther from station T than it is from station S. Find the equation for the hyperbola on which the ship is located. A. C. B. D. Example 6

End of the Lesson