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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
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
Determine the eccentricity of the ellipse given by
A. 0.632
B. 0.775
C. 0.845
D. 1.290
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
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.
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 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.
Graph a Hyperbola
Divide each side
by 60.
The equation is now in standard form with h = –3,
Graph a Hyperbola
Graph the center, vertices, foci, and asymptotes.
Then make a table of values to sketch the 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 segment
between 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 segment
between vertices
b =3
Conjugate axis = 2b
a =6
Distance from each vertex
to center
Write Equations Given Characteristics
Answer:
Write an equation for the hyperbola with foci at
(13, –3) and (–5, –3) and transverse 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
Eccentricity equation
Simplify.
The eccentricity of the hyperbola is about 1.33.
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
1752 = 402 + b2
29,025 = b2
Equation relating a, b, and c
c = 175 and a = 40
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
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.
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.
B.
C.
D.