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

Over Lesson 7-2

Graph the ellipse given by 4x 2 + y

2 + 16x – 6y – 39 = 0.

A. B.

C. D.

Over Lesson 7-2

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.

Over Lesson 7-2

A. 0.632

B. 0.775

C. 0.845

D. 1.290

Determine the eccentricity of the ellipse given by

Over Lesson 7-2

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

Over Lesson 7-2

Identify the conic section represented by 8x

2 + 5y 2 – x + 6y = 0.

A. circle

B. ellipse

C. parabola

D. none of the above

You analyzed and graphed ellipses and circles. (Lesson 7-2)

• Analyze and graph equations of hyperbolas.

• Use equations to identify types of conic sections.

• hyperbola

• transverse axis

• conjugate axis

Graph Hyperbolas in Standard Form

A. Graph the hyperbola given by

The equation is in standard form with h = 0, k = 0,

Graph Hyperbolas in Standard Form

Graph the center, vertices, foci, and asymptotes. Then make a table of values to sketch the hyperbola.

Answer:

Graph Hyperbolas in Standard Form

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 cfor a hyperbola

c2 = 4 + 9 a2 = 4 and b2 = 9

B. Graph the hyperbola given by

Solve for c.

Graph Hyperbolas in Standard Form

Use h, k, a, b, and c to determine the characteristics of the hyperbola.

Graph Hyperbolas in Standard Form

Graph the center, vertices, foci, and asymptotes. Then make a table of values to sketch the hyperbola.

Answer:

Graph the hyperbola given by

A. B.

C. D.

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 grouplike terms.

4(x2 + 6x) – (y2 – 4y) = 28 Factor.

4(x2 + 6x + 9) – (y2 – 4y + 4) = 28 + 4(9) – 4Complete thesquares.

4(x + 3)2 – (y – 2)2 = 60 Factor andsimplify.

Divide each side by 60.

Graph a Hyperbola

The equation is now in standard form with h = –3,

Graph the center, vertices, foci, and asymptotes. Then make a table of values to sketch the hyperbola.

Graph a Hyperbola

Answer:

Graph the hyperbola given by 3x2 – y2 – 30x – 4y = –119.

A. B.

C. D.

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 segmentbetween foci

a = 2 Transverse axis = 2a

c = 3 Distance from each focus to center

c2 = a2 + b2

Write Equations Given Characteristics

Answer:

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 segmentbetween vertices

b = 3 Conjugate axis = 2b

a = 6 Distance from each vertexto center

Write Equations Given Characteristics

Answer:

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.

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.

Find the Eccentricity of a Hyperbola

The eccentricity of the hyperbola is about 1.33.

Simplify.

Eccentricity equation

Answer: 1.33

A. 0.59

B. 0.93

C. 1.24

D. 1.69

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

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

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

Use the discriminant to identify the conic section given by 15 + 6y + y2 = –14x – 3x2.

A. ellipse

B. circle

C. hyperbola

D. parabola

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.

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.

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.

Apply Hyperbolas

Apply Hyperbolas

Answer:

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.

Apply Hyperbolas

Original equation

y = 125

Because the ship is 125 miles from the shore, y = 125. Substitute the value of y into the equation and solve for x.

Solve.

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)

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.