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8.6 The Hyperbola • MHR 637
Some ships navigate using a radio navigation system called LORAN, which is an acronym for LOng RAnge Navigation. A ship receives radio signals from pairs of transmitting stations that send signals at the same time. The LORAN equipment detects the difference in the arrival times of the signals and uses the locus definition of the hyperbola to determine the ship’s location.
To determine the equations of hyperbolas, the absolute values of numbers are used. The absolute value of a real number is its distance from zero on a real number line. For a real number represented by x , the absolute
value is written | x |, which means the
positive value, or magnitude, of x.
The diagram shows that the absolute value of −3 is 3 and the absolute value of 3 is 3.
A hyperbola is the set or locus of points P in the plane such that the absolute value of the difference of the distances from P to two fixed points F 1 and F 2 is a constant. F 1 P − F 2 P = k The two fixed points, F 1 and F 2 , are called the foci of the hyperbola. The line segments F 1 P and F 2 P are called the focal radii of the hyperbola.
- 4 – 3 – 2 – 1 0 1 2 3 4 3 units 3 units
|– 3 | (^) = 3 | 3 | (^) = 3
Web Connection www.school.mcgrawhill.ca/resources/ To learn more about LORAN, visit the above web site. Go to Math Resources , then to MATHEMATICS 11 , to find out where to go next. Write a brief report about the origins of LORAN.
8.6 The Hyperbola
638 MHR • Chapter 8
You will need two clear plastic rulers, a sheet of paper, and a pencil. Step 1 Draw a 10-cm line segment near the centre of the piece of paper. Label the two endpoints F (^1) and F 2. Step 2 Choose a length, k centimetres, which is less than the length F 1 F 2. For this investigation, use k = 4 cm. Step 3 Choose a pair of lengths, a centimetres and b centimetres, such that the
absolute value of their difference equals 4, that is, | a − b | = 4. For example, choose
a = 9 cm and b = 5 cm, or a = 5 cm and b = 9 cm, since |9 − 5|= 4 and |5 − 9| = 4. Step 4 Use both rulers to mark two points that are 9 cm from F 1 and 5 cm from F 2. Then, use both rulers to mark two points that are 5 cm from F 1 and 9 cm from F 2.
Step 5 Repeat steps 3 and 4 using different values of a and b , such that | a − b | = 4,
until you have marked enough points to define two curves. Step 6 Draw a smooth curve through each set of points. The two curves form a hyperbola.
- How many axes of symmetry does the hyperbola have?
- In relation to F 1 and F 2 , where is the point of intersection of the axes of symmetry?
- In Step 4, you marked four points for the chosen values of a and b. Are there any values of a and b for which only two points can be marked? If so, what are the values of a and b? Describe the locations of the points on the hyperbola.
- The vertices of this hyperbola are located on F 1 F 2. How is the distance between the vertices of the two curves related to the value of k? Explain.
I NVESTIGATE & I NQUIRE
F 1 F (^2)
F 1 F (^2)
9 cm (^) 5 cm
9 cm 5 cm
F 1 F (^2)
5 cm 9 cm
5 cm (^) 9 cm
640 MHR • Chapter 8
The hyperbola in Example 1 can be modelled graphically, as shown. The hyperbola has two axes of symmetry. The points (−4, 0) and (4, 0) lie on one axis of symmetry. The line segment joining these two points is called the transverse axis. In this case, the length of the transverse axis is 8. The endpoints of the transverse axis are the vertices of the hyperbola.
The points (0, −3) and (0, 3) lie on the other axis of symmetry. The line segment joining these two points is called the conjugate axis. In this case, the length of the conjugate axis is 6. The endpoints of the conjugate axis are called the co-vertices of the hyperbola.
The point of intersection of the transverse axis and the conjugate axis is called the centre of the hyperbola. In this case, the centre is the origin, (0, 0).
The lines x = −4 and x = 4 form a rectangle with the lines y = −3 and y = 3. The graph of the hyperbola lies between the diagonals of this rectangle. The diagonals of this rectangle are called the asymptotes of the hyperbola. These are the lines that the hyperbola approaches for large values of x and y.
The equation of the hyperbola from Example 1 can be written as
− = 1
In this form of the equation, notice that 4 is half the length of the transverse axis, or half the difference between the focal radii, and 3 is half the length of the conjugate axis. Notice also that the equations of the asymptotes are
y = (^) ^3 4
x and y = – ^3 4
x.
The coordinates of the foci are (−5, 0) and (5, 0). Notice that 4 2 + 3 2 = 5 2.
y^2 32
^ x^2 42
x
y
(0, 3)
F 2 (5, 0)
(0, – 3)
0
F 1 (–5, 0) (^) (–4, 0) (4, 0)
y = – 3 – x^ y^ =^3 – 4 x 4
2
4
- 2
- 4
- 6
- 8 – 6 – 4 – 2 2 4 6 8
The standard form of the equation of a hyperbola centred at the origin, with the transverse axis along the x -axis and the conjugate axis along the y -axis, is
− = 1
- The length of transverse axis along the x -axis is 2 a.
- The length of conjugate axis along the y -axis is 2 b.
- The vertices are V 1 (− a , 0) and V 2 ( a , 0).
- The co-vertices are (0, − b ) and (0, b ).
- The foci are F 1 (− c , 0) and F 2 ( c , 0), which are on the transverse axis.
- The equations of the asymptotes are
y = (^) ^ b a^ ^
x and y = – (^) ^ b a^ ^
x.
y^2 b^2
^ x^2 a^2
The standard form of the equation of a hyperbola centred at the origin, with the transverse axis along the y -axis and the conjugate axis along the x -axis, is
− = 1
- The length of transverse axis along the y -axis is 2 a.
- The length of conjugate axis along the x -axis is 2 b.
- The vertices are V 1 (0, − a ) and V 2 (0, a ).
- The co-vertices are (− b , 0) and ( b , 0).
- The foci are F 1 (0, − c ) and F 2 (0, c ), which are on the transverse axis.
- The equations of the asymptotes are y = (^) b
a x and y = – b
a x.
^ x^2 b^2
y^2 a^2
x
y
(0, b )
F 2 ( c , 0)
axis
(0, – b )
transverse
conjugate
axis
0
F 1 (– c , 0 ) V 1 (– a , 0) V 2 ( a , 0)
x
y
F 2 (0, c )
axis
F 1 (0, – c )
transverse
conjugate
axis
V 1 (0, – a )
V 2 (0, a )
0
(– b , 0) (^) ( b , 0)
8.6 The Hyperbola • MHR 641
8.6 The Hyperbola • MHR 643
A hyperbola may not be centred at the origin. As in the case of the circle and the ellipse, a hyperbola can have a centre (h, k ). The translation rules that apply to the circle and the ellipse also apply to the hyperbola.
The standard form of the equation of a hyperbola centred at (h, k ), with the transverse axis parallel to the x -axis and the conjugate axis parallel to the y -axis, is
− = 1
- The length of the transverse axis parallel to the x -axis is 2 a.
- The length of the conjugate axis parallel to the y -axis is 2 b.
- The vertices are V 1 (h − a , k ) and V 2 (h + a , k ).
- The co-vertices are (h, k − b ) and (h, k + b ).
- The foci are F 1 (h − c , k ) and F 2 (h + c , k ), which are on the transverse axis.
- a^2 + b^2 = c^2
The standard form of the equation of a hyperbola centred at (h, k ), with the transverse axis parallel to the y -axis, and conjugate axis parallel to the x -axis, is
− = 1
- The length of the transverse axis parallel to the y -axis is 2 a.
- The length of the conjugate axis parallel to the x -axis is 2 b.
- The vertices are V 1 (h, k − a ) and V 2 (h, k + a ).
- The co-vertices are (h − b , k ) and (h + b , k ).
- The foci are F 1 (h, k − c ) and F 2 (h, k + c ), which are on the transverse axis.
- a^2 + b^2 = c^2
( x − h)^2 b^2
( y − k )^2 a^2
( y − k )^2 b^2
( x – h)^2 a^2
x
y
F 1 ( h – c, k )
0
( h , k – b )
V 1 ( h – a, k )
transverse
conjugate
axis
axis
h
F 2 ( h + c , k )
h – c h – a h + a h + c
( h , k )
k + b ( h ,^ k^ +^ b )
k
k – b
V 2 ( h + a, k )
x
y
0
transverse
conjugate
axis axis
h
F 2 ( h , k + c )
k (^) ( h + b, k ) ( h – b, k )
F 1 ( h , k – c )
h – b h + c
k + a
k – c
k – a
k + c
( h, k )
V 1 ( h, k – a )
V 2 ( h, k + a )
644 MHR • Chapter 8
EXAMPLE 3 Sketching the Graph of a Hyperbola With Centre ( h , k )
Sketch the graph of the hyperbola − = 1. Label the foci.
SOLUTION
The centre is C(h, k ) = (2, −1).
Since the equation is in the form − = 1, the transverse axis
is parallel to the x -axis. a^2 = 36, so a = 6 b^2 = 16, so b = 4 The length of the transverse axis is 2 a , or 12. The length of the conjugate axis is 2 b , or 8.
The vertices are V 1 (h − a , k ) and V 2 (h + a , k ). Substitute the values of h, k , and a. The vertices are V 1 (2 − 6, −1) and V 2 (2 + 6, −1), or V 1 (−4, −1) and V 2 (8, −1).
The co-vertices are (h, k − b ) and (h, k + b ). Substitute the values of h, k , and b. The co-vertices are (2, − 1 − 4) and (2, − 1 + 4), or (2, −5) and (2, 3).
The foci are F 1 (h − c , k ) and F 2 (h + c , k ). To find c , use a^2 + b^2 = c^2 , with a = 6 and b = 4.
c^2 = 6 2 + 4 2 = 36 + 16 = 52 c = 52 = 2 13
The coordinates of the foci are (2 − 2 13 , −1) and (2 + 2 13 , −1), or approximately (−5.21, −1) and (9.21, −1).
To sketch the graph of the hyperbola, plot the vertices and the co-vertices. Then, construct the rectangle that goes through all four of these points. Sketch the asymptotes by extending the diagonals of the rectangle.
( y − k )^2 b^2
( x^ −^ h)^2 a^2
( y + 1) 2 16
( x^ −^ 2)^2 36
646 MHR • Chapter 8
The equation of the hyperbola is − = 1.
The foci are F 1 (h, k − c ) and F 2 (h, k + c ). To find c , use a^2 + b^2 = c^2 , with a = 5 and b = 2. c^2 = a^2 + b^2 = 5 2 + 2 2 = 25 + 4 = 29 c = 29
The foci are located at (h, k − c ) and (h, k + c ). Thus, the coordinates of the foci are (2, 3 − 29 ) and (2, 3 + 29 ), or approximately (2, −2.39) and (2, 8.39).
To sketch the graph of the hyperbola, plot the vertices and the co-vertices. Then, construct the rectangle that goes through all four of these points. Sketch the asymptotes by extending the diagonals of the rectangle.
Since the transverse axis is parallel to the y -axis, the hyperbola must open up and down. To sketch a branch of the hyperbola, start at a vertex and approach the asymptotes. Sketch the other branch in the same way. Label the foci.
Note that hyperbolas can be graphed using a graphing calculator. As with circles and ellipses, the equations of hyperbolas must first be solved for y.
For example, solving − = 1, results in y = ±^2 3
9 + x^2.
Enter both of the resulting equations in the Y= editor.
Y 1 = (^) ^2 3
9 + x^2 and Y 2 = – ^2 3
9 + x^2
^ x^2 9
y^2 4
( x^ −^ 2)^2 4
( y − 3)^2 25
y
2
–2 0 2 4 6^ x
4
6
8
10
V 1 (2, –2)
(0, 3) (4, 3)
V 2 (2, 8)
F 1 (2, 3 – 29)
F 2 (2, 3 + 29)
8.6 The Hyperbola • MHR 647
K e y C o n c e p t s
- A hyperbola is the set, or locus, of points P in the plane such that the absolute value of the difference of the distances from P to two fixed points F 1 and F 2 is a constant.
|F 1 P^ −^ F 2 P| =^ k
- The standard form of the equation of a hyperbola, with centre at the origin, is either − = 1 (transverse axis along the x -axis) or − = 1
(transverse axis along the y -axis).
- The standard form of the equation of a hyperbola, centred at (h, k ), is either − = 1 (transverse axis parallel to the x -axis)
or − = 1 (transverse axis parallel to the y -axis).
C o m m u n i c a t e Y o u r U n d e r s t a n d i n g
- Describe how you would use the locus definition of the hyperbola to find an equation of the hyperbola with foci F 1 (0, −5) and F 2 (0, 5), and with the constant difference between the focal radii equal to 8.
- Describe how you would sketch the graph of − = 1.
- Describe how you would write an equation in standard form for the hyperbola with vertices (3, −4) and (3, 6) and co-vertices (1, 1) and (5, 1).
- How would you know if a) an equation in standard form models an ellipse or a hyperbola? b) a graph models an ellipse or a hyperbola?
- Is a hyperbola a function? Explain.
( y + 4) 2 ( x − 2) (^) 9 2 25
( x^ −^ h)^2 b^2
( y − k )^2 a^2
( y − k)^2 b^2
( x^ −^ h)^2 a^2
^ x^2 b^2
y^2 a^2
y^2 b^2
^ x^2 a^2
Adjust the window variables , if necessary, and use the Zsquare instruction.
8.6 The Hyperbola • MHR 649
- For each hyperbola, determine i) the coordinates of the centre ii) the coordinates of the vertices and co-vertices iii) the lengths of the transverse and conjugate axes iv) an equation in standard form v) the coordinates of the foci a)
b)
c)
d)
- Sketch the graph of each hyperbola. Label the coordinates of the centre, the vertices, the co-vertices, and the foci.
a) − = 1
b) − = 1
c) 16 x^2 − 9( y − 2) 2 = 144 d) 4( y − 2) 2 −9( x − 5) 2 = 36
- Determine an equation in standard form for each of the following hyperbolas. a) vertices (−2, −5) and (6, −5), foci (−4, −5) and (8, −5) b) centre (0, 3), one vertex (0, 5), and one focus (0, 6) c) centre (−3, 1), one focus (−5, 1), and length of conjugate axis 4 d) centre (−2, 2), one focus (−2, 5), and length of conjugate axis 6
( x^ −^ 1)^2 25
( y + 2) 2 81
( y + 1) 2 ( x − 3) (^) 9 2 16
y
2
4
6
2
4
6
y
y
2
0 2 4 6^ x
4
6
650 MHR • Chapter 8
Apply, Solve, Communicate
- Marine biology A ship is monitoring the movement of a pod of whales with its radar. The radar screen can be modelled as a coordinate grid with the ship at the centre (0, 0). The pod appears to be moving along a curve such that the absolute value of the difference of its distances from (2, 7) and (2, −3) is always 6. Write an equation in standard form to describe the path of the pod.
- Motion in space Some comets travel on hyperbolic paths and never return. Suppose the hyperbolic path of a comet is modelled on a grid with a scale of 1 unit = 3 000 000 km. The vertex of the path is at (2, 0), and the focus is at (6, 0). Write an equation in standard form to model the path of the comet.
- Conjugate hyperbolas a) Two hyperbolas are centred at (1, 2). One has a transverse axis parallel to the x -axis, and the other has a transverse axis parallel to the y -axis. They share the same pair of asymptotes. The equation
of one hyperbola is − = 1. Find the lengths of the
conjugate and transverse axes, and the equation in standard form, of the other hyperbola. b) The two hyperbolas in part a) are known as conjugate hyperbolas. Write equations in standard form for another pair of conjugate hyperbolas with their common centre not at the origin. Explain your reasoning.
- Application Two park rangers were stationed at separate locations on the same side of a lake. The rangers were 6 km apart. Both rangers saw a lightning bolt strike the ground on the other side of the lake and heard the clap of thunder. a) If one ranger heard the clap of thunder 4 s before the other, write an equation that describes all the possible locations of the thunder clap. Place the two rangers on the y -axis, with the midpoint between the rangers at the origin. The speed of sound is approximately 0.35 km/s. b) Sketch the possible locations where the lightning bolt hit the ground. Include the ranger stations in the sketch.
( x^ −^ 2)^2 9
( y − 1)^2 25
B
652 MHR • Chapter 8
- Equilateral hyperbolas The following are examples of equilateral hyperbolas. i) x^2 − y^2 = 4 ii) x^2 − y^2 = 9 iii) y^2 − x^2 = 16 iv) y^2 − x^2 = 25 a) What do the equations have in common? b) Graph each hyperbola. State the equations of its asymptotes. c) What do the graphs have in common that is different from the other hyperbolas you have graphed?
- Standard form Consider a hyperbola with its transverse axis parallel to the x -axis, foci at (− c , 0) and ( c , 0), and vertices at (− a , 0) and ( a , 0).
a) Use the formula for the length of a line segment, l = ( x 2 − x 1 )^2 + ( y 2 − y 1 )^2 , to show that, for a point P( x , y ) on the hyperbola,
( x − c )^2 + y^2 − ( x + c )^2 + y^2 = 2 a and ( x + c ) 2 + y^2 − ( x − c )^2 + y^2 = 2 a.
b) From the equations in part a), derive − = 1.
c) Derive − = 1, the standard form for the hyperbola centred at
(0, 0) with transverse axis parallel to the x -axis.
- Inverse variation When one variable increases and another variable decreases, such that their product is a constant, the relationship can be represented by xy = k , k ≠ 0. The relationship can be stated verbally as y varies inversely as x. The graph of the inverse variation is one branch of a hyperbola, for x , y > 0. Complete the following using a graphing calculator. a) Graph xy = 2, xy = 5, and xy =10. What are the similarities and differences for xy = k? b) Graph xy = −2, xy = −5, and xy = −10. Compare these graphs with the graphs in part a). c) Explain why the graphs in parts a) and b) do not go through the origin. d) Can you graph xy = 0? Explain.
y^2 b^2
^ x^2 a^2
y^2 b^2
^ x^2 c^2 − b^2
Knowledge/Understanding Thinking/Inquiry/Problem Solving Communication Application Find an equation for the locus of points such that the product of the slopes of the lines from a point on the locus to the points (6, 0) and (−6, 0) is 4.
A C H I E V E M E N T Check
