8.7 The Parabola The Hubble Space Telescope orbits the Earth at an altitude of approimatel 600 km. The telescope takes about ninet minutes to complete one orbit. Since it orbits above the Earth s atmosphere, the telescope can perform its scientific work without the negative effects of the atmosphere. The primar reflector of the Hubble Space Telescope is parabolic. A parabola is the set or locus of points P in the plane such that the distance from P to a fied point F equals the distance from P to a fied line l. Focus F PF = PD The fied point F is called the focus. The fied line l is called the directri. Verte P The verte of a parabola is located midwa between the focus and the directri. The parabola never crosses the directri. We will use this information to help sketch parabolas. Directi D l INVESTIGATE & INQUIRE You will need two clear plastic rulers, a sheet of paper, and a pencil. Step Draw a 5-cm line segment near the bottom of the piece of paper. Label the line l. Mark a point about 4 cm above the middle of the segment. Label this point F. F l 8.7 The Parabola MHR 653
Step 2 Choose a length, k cm, that is greater than or equal to half the distance from the point F to the line l. You ma want to make k less than 8 cm. Step 3 To locate a point P that is k cm from line l and k cm from point F, place one ruler so that its 0 mark is on line l and the ruler is perpendicular to line l. Place the other ruler so that its 0 mark is on point F. Adjust the positions of the rulers to locate P that is k cm from l and k cm from F. k k F l Step 4 Mark a second point that is also k cm from line l and k cm from point F. Step 5 Repeat steps 3 and 4 using different values for k until ou have marked enough points to define a complete curve. Step 6 Draw a smooth curve through the points. The curve is an eample of a parabola.. How man aes of smmetr does the parabola have? 2. Steps 3 and 4 instructed ou to mark two points for the chosen distance, k. Are there an values of k for which onl one point can be marked? If so, describe the location of the point on the parabola. 3. What is the relationship between the verte of the parabola, line l, and point F? 4. In Step 2, wh must k be greater than or equal to half the distance from line l to the point F? 654 MHR Chapter 8
EXAMPLE Finding the Equation of a Parabola From its Locus Definition Use the locus definition of the parabola to find an equation of the parabola with focus F(0, 3) and directri = 3. SOLUTION Draw a diagram. The verte is located midwa between the focus and directri, so V(0, 0) is the verte. Since the parabola never crosses the directri, the parabola must open up. Let P(, ) be an point on the parabola. 2 The focus is F(0, 3). Let D(, 3) be an point on the directri. The locus definition of the parabola can be stated algebraicall as PF = PD. Use the formula for the length of a line segment, l = ( 2 ) 2 + ( 2, ) 2 to rewrite PF and PD. PF = ( 0) 2 + ( 3) 2 = 2 + ( 3) 2 PD = ( ) 2 + ( ( 3)) 2 = ( + 3) 2 Substitute: 2 + ( 3) 2 = ( + 3) Square both sides: 2 + ( 3) 2 = ( + 3) 2 Epand: 2 + 2 6 + 9 = 2 + 6 + 9 Solve for : 2 = 2 2 = 2 An equation of the parabola is = 2. 2 6 4 2 4 2 0 F(0, 3) 2 4 6 V(0, 0) = 3 In Eample, note that, when the focus is (0, 3) and the directri is = 3, the equation of the parabola is = 2, and that 2 = 4 3. 2 In general, if the focus is F(0, p) and the directri is = p, then the equation of the parabola is = 2. 8.7 The Parabola MHR 655
The standard form of the equation of a parabola with its verte at the origin and a horizontal directri is = 2 The verte is V(0, 0). If p > 0, the parabola opens up. If p < 0, the parabola opens down. The focus is F(0, p). The equation of the directri is = p. The ais of smmetr is the -ais. The standard form of the equation of a parabola with its verte at the origin and a vertical directri is = 2 The verte is V(0, 0). If p > 0, the parabola opens right. If p < 0, the parabola opens left. The focus is F(p, 0). The equation of the directri is = p. The ais of smmetr is the -ais. = 4p 2 F(0, p) P(, ) ( p, ) P(, ) 0 V(0, 0) V(0, 0) 0 F(p, 0) = p (, p) = p = 4p 2 EXAMPLE 2 Writing an Equation of a Parabola With Verte (0, 0) Write an equation in standard form for the parabola with focus (4, 0) and directri = 4. SOLUTION The verte is located midwa between the focus and the directri, so the verte is V(0, 0). Since the parabola never crosses the directri, the parabola opens right. 656 MHR Chapter 8
The standard form of the equation of a parabola opening right with verte at the origin is = 2. The directri is = p, so p = 4. Since p = 4, the equation is = 2. 6 The equation can be modelled graphicall. 4 2 V(0, 0) F(4, 0) 4 2 0 2 4 2 = 4 4 = 6 2 EXAMPLE 3 Determining the Characteristics of a Parabola From its Equation For the parabola = 8 2, determine the direction of the opening, the coordinates of the verte and the focus, and the equation of the directri. SOLUTION The equation is in the form =, so the graph opens up or down. Find the value of p. Rewrite = 8 2 as =. 4( 2) So, p = 2. Since p < 0, the graph opens down. 2 V(0, 0) The verte is V(0, 0). The focus is F(0, p), or F(0, 2). The equation of the directri is = p, or = 2. 4 2 0 2 The equation can be modelled graphicall. = 2 2 4 6 F(0, 2) = 2 8 Recall that we can translate a parabola = a 2, with verte at the origin, to = a( h) 2 + k b translating h units to the left or right and k units up or down. This translation results in a parabola with verte (h, k). The equation of the resulting parabola can be epressed in standard form. 8.7 The Parabola MHR 657
The standard form of the equation of a parabola with verte V(h, k) and a horizontal directri is k = ( h) 2 The verte is V(h, k). If p > 0, the parabola opens up. If p < 0, the parabola opens down. The focus is F(h, k + p). The equation of the directri is = k p. The equation of the ais of smmetr is = h. The standard form of the equation of a parabola with verte V(h, k) and a vertical directri is h = ( k) 2 The verte is V(h, k). If p > 0, the parabola opens right. If p < 0, the parabola opens left. The focus is F(h + p, k). The equation of the directri is = h p The equation of the ais of smmetr is = k. = h h = ( k) 2 4p F(h, k + p) V(h, k) k = ( h) 2 4p V(h, k) F(h + p, k) = k = k p 0 0 = h p EXAMPLE 4 Writing an Equation of a Parabola With Verte (h, k) Write an equation in standard form for a parabola with focus F( 2, 6) and directri = 4. SOLUTION Since the parabola never crosses the directri, the parabola opens left. The standard form of the equation is h = ( k) 2. The verte is located midwa between the focus and the directri. The coordinates of the verte are V(, 6). Since the verte is V(, 6), h = and k = 6. 658 MHR Chapter 8
Use the focus to find the value of p. The focus F(h + p, k) is ( 2, 6). So, ( + p, 6) = ( 2, 6) + p = 2 p = 3 Now substitute known values into the standard form of the equation. h = ( k) 2 = ( 6) 2 4( 3) = ( 6) 2 2 The equation can be modelled graphicall. 2 0 = ( 6) 2 2 8 = 4 F( 2, 6) 6 4 V(, 6) 2 6 4 2 0 2 4 A equation of the parabola in standard form is = ( 6) 2. 2 Recall that parabolas can be graphed using a graphing calculator. If the equation of a parabola that opens left or right is given in standard form, first solve the equation for. For eample, solving = 4 2 for results in =± 4. Enter both of the resulting equations in the Y= editor. Y = 4 Y2 = 4. Adjust the window variables if necessar, and use the Zsquare instruction. 8.7 The Parabola MHR 659
Ke Concepts A parabola is the set or locus of points P in the plane such that the distance from P to a fied point F equals the distance from P to a fied line l. PF = PD The fied point F is called the focus. The fied line l is called the directri. The verte of a parabola is located midwa between the focus and the directri. The standard form of a parabola with verte at the origin is = 2 (opens up if p > 0, or down if p < 0), or = 2 (opens right if p > 0, or left if p < 0). The standard form of a parabola with verte (h, k) is k = ( h) 2 (opens up if p > 0, or down if p < 0), or h = ( k) 2 (opens right if p > 0, or left if p < 0). Communicate Your Understanding. In our own words, define the following terms. a) verte b) directri c) focus 2. Describe how ou would use the locus definition to find the equation of a parabola with focus F(2, 0) and directri = 2. 3. Describe the relationship between the ais of smmetr and the directri of an parabola. 4. Describe the similarities and differences between the parabolas 3 = 8 ( + 2) 2 and 3 = 8 ( + 2) 2. 5. Describe how ou would determine an equation in standard form for a parabola with focus F(, 3) and directri =. 660 MHR Chapter 8
Practise A. Use the locus definition of the parabola to write an equation for each of the following parabolas. a) focus (0, 2), directri = 2 b) focus (, 3), directri = 4 2. Determine the coordinates of the verte for each of the following parabolas. a) focus (6, 3), directri = 2 b) focus (3, 0), directri = 3 c) focus ( 4, 2), directri = d) focus ( 3, 4), directri = 2 3. Write an equation in standard form for the parabola with the given focus and directri. Sketch the parabola. a) focus (0, 6), directri = 6 b) focus (0, 4), directri = 4 d) focus ( 8, 0), directri = 8 e) focus (0, 2), directri = 2 f) focus (, 0), directri = g) focus (0, 3), directri = 3 h) focus ( 5, 0), directri = 5 Appl, Solve, Communicate 4. Write an equation in standard form for the parabola with the given focus and directri. Sketch the parabola. a) focus (6, 2), directri = 0 b) focus (0, 4), directri = 5 c) focus (2, 2), directri = 5 d) focus (, 4), directri = 2 e) focus ( 3, 5), directri = 5. For each of the following parabolas, determine the direction of the opening, the coordinates of the verte and the focus, and the equation of the directri. Sketch the graph, and determine the domain and range. a) = 2 b) = 6 8 2 c) = 8 2 d) = 2 6 e) 3 = 4 ( + 2) 2 f) + 2 = ( 5) 2 0 g) = 5 ( + ) 2 h) + 3 = ( 2) 2 2 0 i) 2 = ( + 6) 2 2 6. Headlights Automobile headlights contain a parabolic reflector. A bulb with two filaments is used to produce low and high beams. The filament at the focus of the parabola produces the high beam. Light from the filament at the focus is reflected from the parabolic reflector to produce parallel light ras, projecting the light a greater distance. Suppose the filament for the high beams is 5 cm from the verte of the reflector. Write an equation in standard form that models the parabola. Assume that the verte is at the origin and that the filament is 5 units to the right of the origin on the -ais. 8.7 The Parabola MHR 66
B 8. Parabolic reflector TV technicians use parabolic reflectors to pick up the sounds from the plaing field at sporting events. The reflector focuses the incoming sound waves on a microphone, which is located at the focus of the reflector. Suppose the microphone is located 5 cm from the verte of the reflector. a) Write an equation in standard form for the parabolic reflector. Assume that the verte is at the origin and that the microphone is to the left of the verte on the -ais. b) Find the width of the reflector, to the nearest centimetre, at a horizontal distance of 30 cm from the verte. 9. Application The stream of water from some water fountains follows a path in the form of a parabolic arch. For one fountain, the maimum height of the water is 8 cm, and the horizontal distance of the water flow is 2 cm. (0, 0) a) Find an equation in standard form that models the continuous flow of water. Assume that the water spout is located at the origin. b) At a horizontal distance of 0 cm from the origin, what is the height of the water, to the nearest tenth of a centimetre? 0. Skateboard ramp For a skateboarding competition, the organizers would like to use a parabolic ramp with a depth of 5 m, and a width of 5 m. Assume that the starting point of a skateboarder at the top of the ramp is (0, 0), as shown. a) Find an equation in standard form that models the parabolic ramp. b) Find the depth of the ramp, to the nearest tenth of a metre, at a horizontal distance of 0 m from the origin. (0, 0). Parabolic antenna A parabolic antenna is 320 cm wide at a distance of 50 cm from its verte. Determine the distance of the focus from the verte. 8 cm 2 cm 5 m 5 m 662 MHR Chapter 8
2. Motion in space A spacecraft is in a circular orbit 50 km above Earth. When it reaches the velocit needed to escape the Earth s gravit, the spacecraft will follow a parabolic path with the focus at the centre of the Earth, as shown. Suppose the spacecraft reaches its escape velocit above the North Pole. Assume the radius of the Earth is 6400 km. Write an equation in standard form for the parabolic path of the spacecraft. Circular orbit 0 North Pole Parabolic orbit Earth 3. Hubble Space Telescope The primar reflector of the Hubble Space Telescope is parabolic and has a diameter of 4.27 m and a depth of 0.75 m. a) If a camera is recording pictures at the focus of the reflector, how far is the camera from the verte, to the nearest hundredth of a metre? b) Write an equation in standard form that models the parabolic reflector. Sketch the location of the reflector on the coordinate aes for this equation. 4. Technolog Use a graphing calculator to graph each parabola. a) + 3 = 4 ( 2) 2 b) + = ( 5) 2 2 c) 3 = 8 ( + 2) 2 5. Communication a) Use a graphing calculator to graph the famil of parabolas of the form = 2 for p =, 2, 3. b) Now graph for p = 2, 3, 4. c) How are the graphs alike? How are the different? d) What happens to the parabola as p gets closer to 0? e) What happens to the foci as p gets closer to 0? C 6. Inquir/Problem Solving Use the locus definition of the parabola to derive the equation in standard form for a parabola with verte (0, 0), focus (0, p) and directri = p. 8.7 The Parabola MHR 663
7. Standard form Use the locus definition of the parabola to derive the equation in standard form for a parabola with verte (0, 0), focus (p, 0) and directri = p. A CHIEVEMENT Check Knowledge/Understanding Thinking/Inquir/Problem Solving Communication Application A parabolic bridge is 40 m across and 25 m high. What is the length of a stabilizing beam across the bridge at a height of 6 m? CAREER CONNECTION Communications The need for people to communicate with each other has been an important aspect of human histor. Modern communication between people can take man forms, including travelling to see each other, mailing a letter, or making a phone call. For a countr as large as Canada to compete economicall, a highl developed communications industr is essential. Sending information b electronic means, including radio, television, and the Internet, is an increasingl important aspect of the communications industr.. TV satellite dish A satellite dish picks up TV signals from a satellite. The signals travel in parallel paths. When the signals reach the dish, the are reflected to the focus, where the detector is located. Suppose that the focus is located 20 cm from the verte. a) Find an equation in standard form that models the shape of the satellite dish. Sketch the location of the dish on the coordinate aes for this equation. b) Find the width of the dish 20 cm from the verte. 2. Research Use our research skills to investigate each of the following. a) a career that interests ou in the communications industr, including the education and training required and the tpe of work involved b) the work of Marshall McLuhan (9 980), a Canadian who was world famous for his work on communications and the media 664 MHR Chapter 8