This pdf includes the following topics:- The Parabola Definition of Parabola Standard Form of the Equation of a Parabola Using the Standard Form of the Equation of a Parabola The Latus Rectum and Graphing Parabolas Examples
1. P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 900 900 Chapter 9 Conic Sections and Analytic Geometry 79. Write 4x2 - 6xy + 2y2 - 3x + 10y - 6 = 0 as a quadratic In Exercises 85–88, determine whether each statement is true or false. equation in y and then use the quadratic formula to express y If the statement is false, make the necessary change(s) to produce a in terms of x. Graph the resulting two equations using a true statement. graphing utility in a 3 - 50, 70, 104 by 3- 30, 50, 104 viewing 85. If one branch of a hyperbola is removed from a graph, then rectangle. What effect does the xy-term have on the graph of the branch that remains must define y as a function of x. the resulting hyperbola? What problems would you 86. All points on the asymptotes of a hyperbola also satisfy the encounter if you attempted to write the given equation in hyperbola’s equation. standard form by completing the square? x2 y2 2 87. The graph of - = 1 does not intersect the line y = - x. x2 y2 xƒxƒ yƒyƒ 9 4 3 80. Graph - = 1 and - = 1 in the same viewing 88. Two different hyperbolas can never share the same asymptotes. 16 9 16 9 rectangle. Explain why the graphs are not the same. x2 y2 89. What happens to the shape of the graph of 2 - 2 = 1 as c a b : q , where c2 = a2 + b2? a Critical Thinking Exercises 90. Find the standard form of the equation of the hyperbola with Make Sense? In Exercises 81–84, determine whether each vertices 15, -62 and (5, 6), passing through (0, 9). statement makes sense or does not make sense, and explain your 91. Find the equation of a hyperbola whose asymptotes are reasoning. perpendicular. 81. I changed the addition in an ellipse’s equation to subtraction and this changed its elongation from horizontal to vertical. Preview Exercises Exercises 92–94 will help you prepare for the material covered in 82. I noticed that the definition of a hyperbola closely resembles the next section. that of an ellipse in that it depends on the distances between a set of points in a plane to two fixed points, the foci. In Exercises 92–93, graph each parabola with the given equation. 92. y = x2 + 4x - 5 93. y = - 31x - 122 + 2 83. I graphed a hyperbola centered at the origin that had y-intercepts, but no x-intercepts. 94. Isolate the terms involving y on the left side of the equation: 84. I graphed a hyperbola centered at the origin that was y2 + 2y + 12x - 23 = 0. symmetric with respect to the x-axis and also symmetric Then write the equation in an equivalent form by completing with respect to the y-axis. the square on the left side. Section 9.3 The Parabola Objectives At first glance, this image looks like columns of smoke rising from a ��� Graph parabolas with vertices fire into a starry sky. Those are, indeed, stars in the background, but you are not looking at ordinary smoke columns. These stand almost at the origin. 6 trillion miles high and are 7000 light-years from Earth—more ��� Write equations of parabolas than 400 million times as far away as the sun. in standard form. ��� Graph parabolas with vertices not at the origin. T his NASA photograph is one of a series of stunning images captured from the ends of the universe by the Hubble Space ��� Solve applied problems Telescope. The image shows infant involving parabolas. star systems the size of our solar system emerging from the gas and dust that shrouded their creation. Using a parabolic mirror that is 94.5 inches in diameter, the Hubble has provided answers to many of the profound mysteries of the cosmos: How big and how old is the universe? How did the galaxies come to exist? Do other Earth-like planets orbit other sun-like stars? In this section, we study parabolas and their applications, including parabolic shapes that gather distant rays of light and focus them into spectacular images. Definition of a Parabola In Chapter 2, we studied parabolas, viewing them as graphs of quadratic functions in the form y = a1x - h22 + k or y = ax2 + bx + c.
2. P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 901 Section 9.3 The Parabola 901 Study Tip Here is a summary of what you should already know about graphing parabolas. Graphing y ⴝ a(x ⴚ h)2 ⴙ k and y ⴝ ax 2 ⴙ bx ⴙ c 1. If a 7 0, the graph opens upward. If a 6 0, the graph opens downward. 2. The vertex of y = a1x - h22 + k is 1h, k2. y y x=h y = a(x − h)2 + k (h, k) a<0 x x y = a(x − h)2 + k a>0 (h, k) x=h b 3. The x-coordinate of the vertex of y = ax2 + bx + c is x = - . 2a Parabola Parabolas can be given a geometric definition that enables us to include graphs that open to the left or to the right, as well as those that open obliquely. The Directrix definitions of ellipses and hyperbolas involved two fixed points, the foci. By Axis of contrast, the definition of a parabola is based on one point and a line. Focus symmetry Vertex Definition of a Parabola A parabola is the set of all points in a plane that are equidistant from a fixed line, the directrix, and a fixed point, the focus, that is not on the line (see Figure 9.29). Figure 9.29 In Figure 9.29, find the line passing through the focus and perpendicular to the directrix. This is the axis of symmetry of the parabola. The point of intersection of the parabola with its axis of symmetry is called the vertex. Notice that the vertex is midway between the focus and the directrix. Standard Form of the Equation of a Parabola The rectangular coordinate system enables us y to translate a parabola’s geometric definition d1 M(−p, y) into an algebraic equation. Figure 9.30 is our P(x, y) starting point for obtaining an equation. We d2 place the focus on the x-axis at the point 1p, 02. The directrix has an equation given by x x = - p. The vertex, located midway between the focus and the directrix, is at the origin. Focus (p, 0) Directrix: x = −p What does the definition of a parabola tell us about the point 1x, y2 in Figure 9.30? For any point 1x, y2 on the parabola, the distance d1 to the directrix is equal to the distance d2 to the focus. Thus, the point 1x, y2 Figure 9.30 is on the parabola if and only if d1 = d2 . 41x + p2 + 1y - y2 = 41x - p2 + 1y - 02 2 2 2 2 Use the distance formula. 1x + p22 = 1x - p22 + y2 Square both sides of the equation.
3. P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 902 902 Chapter 9 Conic Sections and Analytic Geometry x2 + 2px + p2 = x2 - 2px + p2 + y2 Square x + p and x - p. 2px = - 2px + y2 Subtract x 2 + p 2 from both sides of the equation. y2 = 4px Solve for y 2. This last equation is called the standard form of the equation of a parabola with its vertex at the origin. There are two such equations, one for a focus on the x-axis and one for a focus on the y-axis. Standard Forms of the Equations of a Parabola The standard form of the equation of a parabola with vertex at the origin is y 2 = 4px or x2 = 4py. Figure 9.31(a) illustrates that for the equation on the left, the focus is on the x-axis, which is the axis of symmetry. Figure 9.31(b) illustrates that for the equation on the right, the focus is on the y-axis, which is the axis of symmetry. y y x2 = 4py y2 = 4px Focus (0, p) Vertex Vertex x x Study Tip Focus (p, 0) It is helpful to think of p as the Directrix: x = −p directed distance from the vertex to Directrix: y = −p the focus. If p 7 0, the focus lies p units to the right of the vertex or p units above the vertex. If p 6 0, the focus lies ƒ p ƒ units to the left of the Figure 9.31(a) Parabola with the x-axis as the Figure 9.31(b) Parabola with the y-axis axis of symmetry. If p 7 0, the graph opens to the as the axis of symmetry. If p 7 0, the graph vertex or ƒ p ƒ units below the vertex. right. If p 6 0, the graph opens to the left. opens upward. If p 6 0, the graph opens downward. ��� Graph parabolas with vertices at Using the Standard Form of the Equation of a Parabola the origin. We can use the standard form of the equation of a parabola to find its focus and directrix. Observing the graph’s symmetry from its equation is helpful in locating the focus. y2=4px x2=4py The equation does not change if The equation does not change if y is replaced with −y. There is x is replaced with −x. There is x-axis symmetry and the focus is y-axis symmetry and the focus is on the x-axis at (p, 0). on the y-axis at (0, p). Although the definition of a parabola is given in terms of its focus and its directrix, the focus and directrix are not part of the graph. The vertex, located at the origin, is a point on the graph of y2 = 4px and x2 = 4py. Example 1 illustrates how you can find two additional points on the parabola. EXAMPLE 1 Finding the Focus and Directrix of a Parabola Find the focus and directrix of the parabola given by y 2 = 12x. Then graph the parabola.
4. P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 903 Section 9.3 The Parabola 903 y Solution The given equation, y2 = 12x, is in the standard form y2 = 4px, 7 so 4p = 12. 6 (3, 6) No change if y is Directrix: x = −3 5 replaced with −y. 4 The parabola has y2=12x 3 x-axis symmetry. 2 Focus (3, 0) This is 4p. 1 x −5 −4 −2 −1−1 1 2 3 4 5 We can find both the focus and the directrix by finding p. −2 Vertex (0, 0) 4p = 12 −3 −4 p = 3 Divide both sides by 4. −5 (3, −6) Because p is positive, the parabola, with its x-axis symmetry, opens to the right. The −6 −7 focus is 3 units to the right of the vertex, (0, 0). Figure 9.32 The graph of y2 = 12x Focus: 1p, 02 = 13, 02 Directrix: x = - p; x = - 3 Technology The focus, (3, 0), and directrix, x = - 3, are shown in Figure 9.32. We graph y2 = 12x with a graphing To graph the parabola, we will use two points on the graph that lie directly utility by first solving for y. The above and below the focus. Because the focus is at (3, 0), substitute 3 for x in the screen shows the graphs of parabola’s equation, y 2 = 12x. y = 212x and y = - 212x. The graph fails the vertical line test. y2 = 12 # 3 Replace x with 3 in y 2 = 12x. Because y2 = 12x is not a function, y2 = 36 Simplify. you were not familiar with this form y = ; 236 = ; 6 Apply the square root property. of the parabola’s equation in Chapter 2. The points on the parabola above and below the focus are (3, 6) and 13, - 62. The graph is sketched in Figure 9.32. y1 = 兹12x Check Point 1 Find the focus and directrix of the parabola given by y2 = 8x. Then graph the parabola. In general, the points on a parabola y2 = 4px that lie above and below the focus, y2 = −兹12x 1p, 02, are each at a distance ƒ 2p ƒ from the focus. This is because if x = p, then y2 = 4px = 4p2, so y = ; 2p. The line segment joining these two points is called the latus rectum; its length is ƒ 4p ƒ . [−6, 6, 1] by [−8, 8, 1] The Latus Rectum and Graphing Parabolas The latus rectum of a parabola is a line segment that passes through its focus, is parallel to its directrix, and has its endpoints on the parabola. Figure 9.33 shows that the length of the latus rectum for the graphs of y2 = 4px and x2 = 4py is ƒ 4p ƒ . y y x2 = 4py Focus (0, p) y2 = 4px Latus rectum length: ⎥ 4p⎥ x x Focus (p, 0) Directrix: x = −p Latus rectum Directrix: y = −p length: ⎥ 4p⎥ Figure 9.33 Endpoints of the latus rectum are helpful in determining a parabola’s “width,” or how it opens.
5. P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 904 904 Chapter 9 Conic Sections and Analytic Geometry y EXAMPLE 2 Finding the Focus and Directrix of a Parabola 5 4 Find the focus and directrix of the parabola given by x2 = - 8y. Then graph the Directrix: y = 2 3 parabola. Vertex (0, 0) 1 Latus Rectum Solution The given equation, x2 = - 8y, is in the standard form x2 = 4py, so x 4p = - 8. −5 −4 −3 −2 −1−1 1 2 3 4 5 (−4, −2) (4, −2) No change if x is −2 replaced with −x. −3 The parabola has x2=–8y −4 Focus (0, −2) y-axis symmetry. −5 This is 4p. Figure 9.34 The graph of We can find both the focus and the directrix by finding p. x2 = - 8y 4p = - 8 p = -2 Divide both sides by 4. Technology Because p is negative, the parabola, with its y-axis symmetry, opens downward. The 2 Graph x = - 8y by first solving for focus is 2 units below the vertex, (0, 0). x2 y: y = - . The graph passes the Focus: 10, p2 = 10, - 22 8 Directrix: y = - p; y = 2 vertical line test. Because x 2 = - 8y is a function, you were familiar with The focus and directrix are shown in Figure 9.34. the parabola’s alternate algebraic To graph the parabola, we will use the vertex, (0, 0), and the two endpoints of 1 the latus rectum. The length of the latus rectum is form, y = - x2, in Chapter 2. 8 The form is y = ax2 + bx + c, with ƒ 4p ƒ = ƒ 41-22 ƒ = ƒ - 8 ƒ = 8. 1 a = - , b = 0, and c = 0. 8 Because the graph has y-axis symmetry, the latus rectum extends 4 units to the left and 4 units to the right of the focus, 10, -22. The endpoints of the latus rectum are 1- 4, -22 and 14, -22. Passing a smooth curve through the vertex and these two points, we sketch the parabola, shown in Figure 9.34. Check Point 2 Find the focus and directrix of the parabola given by x2 = - 12y. Then graph the parabola. [−6, 6, 1] by [−6, 6, 1] In Examples 1 and 2, we used the equation of a parabola to find its focus and directrix. In the next example, we reverse this procedure. ��� Write equations of parabolas in EXAMPLE 3 Finding the Equation of a Parabola from Its Focus standard form. and Directrix y Find the standard form of the equation of a parabola with focus (5, 0) and directrix x = - 5, shown in 7 6 Figure 9.35. Directrix: 5 Solution The focus is (5, 0). Thus, the focus is x = −5 4 3 on the x-axis. We use the standard form of the 2 Focus (5, 0) equation in which there is x-axis symmetry, namely 1 y2 = 4px. x −4 −3 −2 −1−1 1 2 3 4 5 We need to determine the value of p. Figure 9.35 shows that the focus is 5 units to the −2 −3 right of the vertex, (0, 0). Thus, p is positive and −4 p = 5. We substitute 5 for p in y 2 = 4px to obtain −5 the standard form of the equation of the parabola. −6 The equation is −7 y2 = 4 # 5x or y2 = 20x. Figure 9.35
6. P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 905 Section 9.3 The Parabola 905 Check Point 3 Find the standard form of the equation of a parabola with focus (8, 0) and directrix x = - 8. ��� Graph parabolas with vertices Translations of Parabolas The graph of a parabola can have its vertex at 1h, k2, rather than at the origin. not at the origin. Horizontal and vertical translations are accomplished by replacing x with x - h and y with y - k in the standard form of the parabola’s equation. Table 9.3 gives the standard forms of equations of parabolas with vertex at 1h, k2. Figure 9.36 shows their graphs. Table 9.3 Standard Forms of Equations of Parabolas with Vertex at (h, k) Equation Vertex Axis of Symmetry Focus Directrix Description 1y - k22 = 4p1x - h2 1h, k2 1h + p, k2 If p 7 0, opens to the right. Horizontal x = h - p If p 6 0, opens to the left. 1x - h22 = 4p1y - k2 1h, k2 1h, k + p2 If p 7 0, opens upward. Vertical y = k - p If p 6 0, opens downward. y y (y − k)2 = 4p(x − h) (x − h)2 = 4p(y − k) Focus Directrix: x = h − p (h, k + p) Study Tip If y is the squared term, there is Vertex (h, k) horizontal symmetry and the Vertex (h, k) parabola’s equation is not a function. x x If x is the squared term, there is Focus (h + p, k) vertical symmetry and the parabola’s equation is a function. Continue to Directrix: y = k − p think of p as the directed distance from the vertex, 1h, k2, to the focus. Figure 9.36 Graphs of parabolas with vertex at 1h, k2 and p 7 0 The two parabolas shown in Figure 9.36 illustrate standard forms of equations for p 7 0. If p 6 0, a parabola with a horizontal axis of symmetry will open to the left and the focus will lie to the left of the directrix. If p 6 0, a parabola with a vertical axis of symmetry will open downward and the focus will lie below the directrix. EXAMPLE 4 Graphing a Parabola with Vertex at 1h, k2 Find the vertex, focus, and directrix of the parabola given by 1x - 322 = 81y + 12. Then graph the parabola. Solution In order to find the focus and directrix, we need to know the vertex. In the standard forms of equations with vertex at 1h, k2, h is the number subtracted from x and k is the number subtracted from y. (x-3)2=8(y-(–1)) This is (x − h)2, This is y − k, with h = 3. with k = −1. We see that h = 3 and k = - 1. Thus, the vertex of the parabola is 1h, k2 = 13, -12.
7. P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 906 906 Chapter 9 Conic Sections and Analytic Geometry y Now that we have the vertex, 13, - 12, we can find both the focus and directrix 5 by finding p. The equation is in the standard form 1x - h22 = 4p1y - k2. 4 Latus Rectum 3 (x-3)2=8(y+1) 2 Focus (3, 1) Because x is the squared term, there is vertical symmetry and (−1, 1) (7, 1) This is 4p. the parabola’s equation is a function. x −2 −1 −1 1 2 4 5 6 7 8 Because 4p = 8, p = 2. Based on the standard form of the equation, the axis of −2 Vertex (3, −1) symmetry is vertical. With a positive value for p and a vertical axis of symmetry, the −4 Directrix: y = −3 parabola opens upward. Because p = 2, the focus is located 2 units above the −5 vertex, 13, - 12. Likewise, the directrix is located 2 units below the vertex. Figure 9.37 The graph of Focus: (h, k+p)=(3, –1+2)=(3, 1) 1x - 32 = 81y + 12 2 The vertex, (h, k), The focus is 2 units is (3, −1). above the vertex, (3, −1). Technology Directrix: y=k-p Graph 1x - 32 = 81y + 12 by first 2 y=–1-2=–3 solving for y: 8 1x 1 The directrix is 2 units - 322 = y + 1 below the vertex, (3, −1). y = 181x - 322 - 1. Thus, the focus is (3, 1) and the directrix is y = - 3. They are shown in Figure 9.37. To graph the parabola, we will use the vertex, 13, -12, and the two endpoints of the The graph passes the vertical line test. latus rectum. The length of the latus rectum is Because 1x - 322 = 81y + 12 is a function, you were familiar with the ƒ 4p ƒ = ƒ 4 # 2 ƒ = ƒ 8 ƒ = 8. parabola’s alternate algebraic form, y = 181x - 322 - 1, in Chapter 2. The Because the graph has vertical symmetry, the latus rectum extends 4 units to the left 2 and 4 units to the right of the focus, (3, 1). The endpoints of the latus rectum are 13 - 4, 12, or 1- 1, 12, and 13 + 4, 12, or (7, 1). Passing a smooth curve through the form is y = a1x - h2 + k with 1 a = 8 , h = 3, and k = - 1. vertex and these two points, we sketch the parabola, shown in Figure 9.37. Check Point 4 Find the vertex, focus, and directrix of the parabola given by 1x - 222 = 41y + 12. Then graph the parabola. In some cases, we need to convert the equation of a parabola to standard form by completing the square on x or y, whichever variable is squared. Let’s see how this is done. [−3, 9, 1] by [−6, 6, 1] EXAMPLE 5 Graphing a Parabola with Vertex at 1h, k2 Find the vertex, focus, and directrix of the parabola given by y2 + 2y + 12x - 23 = 0. Then graph the parabola. Solution We convert the given equation to standard form by completing the square on the variable y. We isolate the terms involving y on the left side. y2 + 2y + 12x - 23 = 0 This is the given equation. y2 + 2y = - 12x + 23 Isolate the terms involving y. y2 + 2y + 1 = - 12x + 23 + 1 Complete the square by adding the square of half the coefficient of y. 1y + 122 = - 12x + 24 Factor. To express the equation 1y + 122 = - 12x + 24 in the standard form 1y - k22 = 4p1x - h2, we factor out - 12 on the right. The standard form of the parabola’s equation is 1y + 122 = - 121x - 22.
8. P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 907 Section 9.3 The Parabola 907 Directrix: x = 5 We use 1y + 122 = - 12(x - 2) to identify the vertex, 1h, k2, and the value for p y needed to locate the focus and the directrix. 5 (−1, 5) 4 [y-(–1)]2=–12(x-2) The equation is in the standard form 1y - k22 = 4p1x - h2. Because y is the squared Latus 3 Rectum 2 This is (y − k)2, This is This is x − h, term, there is horizontal symmetry and the parabola’s 1 with k = −1. 4p. with h = 2. x equation is not a function. −3 −2 1 3 4 6 7 −2 We see that h = 2 and k = - 1. Thus, the vertex of the parabola is 1h, k2 = 12, -12. Focus −3 Because 4p = - 12, p = - 3. Based on the standard form of the equation, the axis of (−1, −1) −4 Vertex (2, −1) symmetry is horizontal. With a negative value for p and a horizontal axis of symmetry, −5 the parabola opens to the left. Because p = - 3, the focus is located 3 units to the left of the vertex, 12, - 12. Likewise, the directrix is located 3 units to the right of the −6 (−1, −7) −7 vertex. Figure 9.38 The graph of Focus: y2 + 2y + 12x - 23 = 0, or (h+p, k)=(2+(–3), –1)=(–1, –1) 1y + 122 = - 121x - 22 The vertex, (h, k), The focus is 3 units is (2, −1). to the left of the Technology vertex, (2, −1). Graph y 2 + 2y + 12x - 23 = 0 by solving the equation for y. Directrix: x=h-p x=2-(–3)=5 y2+2y+(12x-23)=0 a=1 b=2 c = 12x − 23 The directrix is 3 units to the right of the Use the quadratic formula to solve for vertex, (2, −1). Thus, the focus is 1-1, - 12 and the directrix is x = 5. They are shown in Figure 9.38. y and enter the resulting equations. -2 + 44 - 4112x - 232 To graph the parabola, we will use the vertex, 12, -12, and the two endpoints y1 = of the latus rectum. The length of the latus rectum is 2 -2 - 44 - 4112x - 232 ƒ 4p ƒ = ƒ 41- 32 ƒ = ƒ -12 ƒ = 12. y2 = 2 Because the graph has horizontal symmetry, the latus rectum extends 6 units above and 6 units below the focus, 1- 1, -12. The endpoints of the latus rectum are y1 1- 1, -1 + 62, or 1- 1, 52, and 1-1, - 1 - 62, or 1-1, -72. Passing a smooth curve through the vertex and these two points, we sketch the parabola shown in Figure 9.38. y2 Check Point 5 Find the vertex, focus, and directrix of the parabola given by y 2 + 2y + 4x - 7 = 0. Then graph the parabola. [−4, 8, 1] by [−8, 6, 1] ��� Solve applied problems involving Applications parabolas. Parabolas have many applications. Cables hung between structures to form suspension bridges form parabolas. Arches constructed of steel and concrete, whose main purpose is strength, are usually parabolic in shape. Parabola Parabola Suspension bridge Arch bridge We have seen that comets in our solar system travel in orbits that are ellipses and hyperbolas. Some comets follow parabolic paths. Only comets with elliptical orbits, such as Halley’s Comet, return to our part of the galaxy.
9. P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 908 908 Chapter 9 Conic Sections and Analytic Geometry If a parabola is rotated about its axis of symmetry, a parabolic surface is formed. Figure 9.39(a) shows how a parabolic surface can be used to reflect light. Light originates at the focus. Note how the light is reflected by the parabolic surface, so that the outgoing light is parallel to the axis of symmetry. The reflective proper- ties of parabolic surfaces are used in the design of searchlights [see Figure 9.39(b)], automobile headlights, and parabolic microphones. The Hubble Space Telescope Outgoing light Axis of symmetry Light at focus Focus Figure 9.39(b) Light from the The Hubble Space Telescope Figure 9.39(a) Parabolic surface focus is reflected parallel to the axis reflecting light of symmetry. For decades, astronomers hoped to create an observatory above the atmosphere that would pro- Figure 9.40(a) shows how a parabolic surface can be used to reflect incoming vide an unobscured view of the light. Note that light rays strike the surface and are reflected to the focus. This universe. This vision was realized principle is used in the design of reflecting telescopes, radar, and television satellite with the 1990 launching of the dishes. Reflecting telescopes magnify the light from distant stars by reflecting the Hubble Space Telescope. The tele- scope initially had blurred vision light from these bodies to the focus of a parabolic mirror [see Figure 9.40(b)]. due to problems with its parabolic mirror. The mirror had been ground two millionths of a meter Eyepiece smaller than design specifications. In 1993, astronauts from the Incoming light Space Shuttle Endeavor equipped Axis of symmetry the telescope with optics to cor- rect the blurred vision. “A small change for a mirror, a giant leap for astronomy,” Christopher J. Focus Parabolic surface Burrows of the Space Telescope Science Institute said when clear images from the ends of the uni- verse were presented to the public Figure 9.40(b) Incoming light rays are Figure 9.40(a) Parabolic surface after the repair mission. reflected to the focus. reflecting incoming light EXAMPLE 6 Using the Reflection Property of Parabolas y An engineer is designing a flashlight using a parabolic (2, 2) reflecting mirror and a light source, shown in Figure 9.41. 2 The casting has a diameter of 4 inches and a depth of 2 inches 2 inches. What is the equation of the parabola used to 1 2 inches shape the mirror? At what point should the light source x be placed relative to the mirror’s vertex? −3 −2 −1 1 2 3 4 inches Solution We position the parabola with its vertex at Figure 9.41 Designing a 4 inches the origin and opening upward (Figure 9.42). Thus, the focus is on the y-axis, located at 10, p2. We use the flashlight Figure 9.42
10. P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 909 Section 9.3 The Parabola 909 standard form of the equation in which there is y-axis symmetry, namely x2 = 4py. We need to find p. Because (2, 2) lies on the parabola, we let x = 2 and y = 2 in x2 = 4py. 2 2 = 4p # 2 Substitute 2 for x and 2 for y in x 2 = 4py. 4 = 8p Simplify. 1 p = 2 Divide both sides of the equation by 8 and reduce the resulting fraction. We substitute 12 for p in x2 = 4py to obtain the standard form of the equation of the parabola. The equation of the parabola used to shape the mirror is x 2 = 4 # 12 y or x2 = 2y. The light source should be placed at the focus, 10, p2. Because p = 12 , the light should be placed at A 0, 12 B , or 12 inch above the vertex. 6 Check Point In Example 6, suppose that the casting has a diameter of 6 inches and a depth of 4 inches. What is the equation of the parabola used to shape the mirror? At what point should the light source be placed relative to the mirror’s vertex? Two intersecting Degenerate Conic Sections Point Line lines We opened the chapter by noting that conic sections are curves that result from the intersection of a cone and a plane. However, these intersections might not result in a conic section. Three degenerate cases occur when the cutting plane passes through the vertex. These degenerate conic sections are a point, a line, and a pair of intersecting lines, illustrated in Figure 9.43. Figure 9.43 Degenerate conics Exercise Set 9.3 Practice Exercises c. d. y y In Exercises 1–4, find the focus and directrix of each parabola with the given equation. Then match each equation to one of the graphs 4 4 that are shown and labeled (a)–(d). 3 3 2 2 1. y 2 = 4x 2. x2 = 4y 1 1 x x 3. x2 = - 4y 4. y 2 = - 4x −4 −3 −2 −1−1 1 2 3 4 −4 −3 −2 −1−1 1 2 3 4 −2 −2 −3 −3 a. b. −4 −4 y y 4 4 In Exercises 5–16, find the focus and directrix of the parabola with 3 3 the given equation. Then graph the parabola. 2 2 5. y2 = 16x 6. y 2 = 4x 1 1 x x 7. y2 = - 8x 8. y 2 = - 12x −4 −3 −2 −1−1 1 2 3 4 −4 −3 −2 −1−1 1 2 3 4 9. x 2 = 12y 10. x2 = 8y −2 −2 11. x2 = - 16y 12. x2 = - 20y −3 −3 −4 −4 13. y2 - 6x = 0 14. x2 - 6y = 0 15. 8x2 + 4y = 0 16. 8y2 + 4x = 0
11. P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 910 910 Chapter 9 Conic Sections and Analytic Geometry In Exercises 17–30, find the standard form of the equation of each In Exercises 43–48, convert each equation to standard form by parabola satisfying the given conditions. completing the square on x or y. Then find the vertex, focus, and 17. Focus: (7, 0); Directrix: x = -7 directrix of the parabola. Finally, graph the parabola. 18. Focus: (9, 0); Directrix: x = -9 43. x2 - 2x - 4y + 9 = 0 44. x2 + 6x + 8y + 1 = 0 19. Focus: 1- 5, 02; Directrix: x = 5 45. y2 - 2y + 12x - 35 = 0 46. y2 - 2y - 8x + 1 = 0 20. Focus: 1- 10, 02; Directrix: x = 10 47. x2 + 6x - 4y + 1 = 0 48. x2 + 8x - 4y + 8 = 0 21. Focus: (0, 15); Directrix: y = - 15 Practice Plus 22. Focus: (0, 20); Directrix: y = - 20 In Exercises 49–54, use the vertex and the direction in which the 23. Focus: 10, - 252; Directrix: y = 25 parabola opens to determine the relation’s domain and range. Is 24. Focus: 10, - 152; Directrix: y = 15 the relation a function? 25. Vertex: 12, - 32; Focus: 12, - 52 49. y2 + 6y - x + 5 = 0 50. y2 - 2y - x - 5 = 0 26. Vertex: 15, - 22; Focus: 17, - 22 51. y = - x2 + 4x - 3 52. y = - x2 - 4x + 4 2 27. Focus: (3, 2); Directrix: x = -1 53. x = - 41y - 12 + 3 54. x = - 31y - 122 - 2 28. Focus: (2, 4); Directrix: x = -4 In Exercises 55–60, find the solution set for each system by 1 - 3, 42; Directrix: graphing both of the system’s equations in the same rectangular 29. Focus: y = 2 coordinate system and finding points of intersection. Check all 30. Focus: 17, - 12; Directrix: y = -9 solutions in both equations. In Exercises 31–34, find the vertex, focus, and directrix of each 1y - 222 = x + 4 1y - 322 = x - 2 parabola with the given equation. Then match each equation to 55. 56. b c 1 x + y = 5 one of the graphs that are shown and labeled (a)–(d). y = - x 31. 1y - 122 = 41x - 12 2 32. 1x + 122 = 41y + 12 x = y2 - 3 x = y2 - 5 57. b 58. b 33. 1x + 122 = - 41y + 12 x = y2 - 3y x2 + y2 = 25 34. 1y - 122 = - 41x - 12 x = 1y + 222 - 1 x = 2y2 + 4y + 5 59. b 60. b a. b. 1x - 22 + 1y + 22 = 1 2 2 1x + 122 + 1y - 222 = 1 y y 4 5 Application Exercises 3 4 61. The reflector of a flashlight is in the shape of a parabolic 2 3 surface. The casting has a diameter of 4 inches and a depth of 1 1 2 inch. How far from the vertex should the light bulb be placed? x 1 −5 −4 −3 −2 −1−1 1 2 3 x 62. The reflector of a flashlight is in the shape of a parabolic −2 −4 −3 −2 −1−1 1 2 3 4 surface. The casting has a diameter of 8 inches and a depth of 1 −3 −2 inch. How far from the vertex should the light bulb be placed? −4 −3 63. A satellite dish, like the one shown below, is in the shape of a c. d. parabolic surface. Signals coming from a satellite strike the surface of the dish and are reflected to the focus, where the y y receiver is located. The satellite dish shown has a diameter of 5 4 12 feet and a depth of 2 feet. How far from the base of the 4 3 dish should the receiver be placed? 3 2 y 2 1 1 x x −5 −4 −3 −2 −1−1 1 2 3 12 feet −4 −3 −2 −1−1 1 2 3 4 −2 Receiver −2 −3 −3 −4 (6, 2) 2 feet x In Exercises 35–42, find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. 35. 1x - 222 = 81y - 12 36. 1x + 222 = 41y + 12 37. 1x + 122 = - 81y + 12 38. 1x + 222 = - 81y + 22 39. 1y + 322 = 121x + 12 40. 1y + 422 = 121x + 22 64. In Exercise 63, if the diameter of the dish is halved and the depth stays the same, how far from the base of the smaller 41. 1y + 122 = - 8x 42. 1y - 122 = - 8x dish should the receiver be placed?
12. P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 911 Section 9.3 The Parabola 911 65. The towers of the Golden Gate Bridge connecting 76. Use a graphing utility to graph any three of the parabolas San Francisco to Marin County are 1280 meters apart and rise that you graphed by hand in Exercises 35–42. First solve the 160 meters above the road. The cable between the towers has given equation for y, possibly using the square root property. the shape of a parabola and the cable just touches the sides of Enter each of the two resulting equations to produce the the road midway between the towers. What is the height of the complete graph. cable 200 meters from a tower? Round to the nearest meter. Use a graphing utility to graph the parabolas in Exercises 77–78. y Write the given equation as a quadratic equation in y and use the Parabolic (640, 160) quadratic formula to solve for y. Enter each of the equations to Cable produce the complete graph. x 77. y2 + 2y - 6x + 13 = 0 78. y2 + 10y - x + 25 = 0 160 meters In Exercises 79–80, write each equation as a quadratic equation in y rs 1280 mete and then use the quadratic formula to express y in terms of x. Graph the resulting two equations using a graphing utility. What effect does the xy-term have on the graph of the resulting parabola? 79. 16x2 - 24xy + 9y2 - 60x - 80y + 100 = 0 66. The towers of a suspension bridge are 800 feet apart and rise 80. x2 + 223xy + 3y2 + 823x - 8y + 32 = 0 160 feet above the road. The cable between the towers has the shape of a parabola and the cable just touches the sides Critical Thinking Exercises of the road midway between the towers. What is the height of the cable 100 feet from a tower? Make Sense? In Exercises 81–84, determine whether each state- ment makes sense or does not make sense, and explain your reasoning. Parabolic (400, 160) 81. I graphed a parabola that opened to the right and contained Cable a maximum point. 82. Knowing that a parabola opening to the right has a vertex at Road 1 - 1, 12 gives me enough information to determine its graph. 800 feet 83. I noticed that depending on the values for A and B, assuming that they are both not zero, the graph of Ax2 + By2 = C can represent any of the conic sections other than a parabola. 67. The parabolic arch shown in the figure is 50 feet above the 84. I’m using a telescope in which light from distant stars is water at the center and 200 feet wide at the base. Will a boat reflected to the focus of a parabolic mirror. that is 30 feet tall clear the arch 30 feet from the center? In Exercises 85–88, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. 85. The parabola whose equation is x = 2y - y2 + 5 opens to 100 ft 100 ft the right. 50 ft 86. If the parabola whose equation is x = ay2 + by + c has its vertex at (3, 2) and a 7 0, then it has no y-intercepts. 87. Some parabolas that open to the right have equations that define y as a function of x. 68. A satellite dish in the shape of a parabolic surface has a 88. The graph of x = a1y - k2 + h is a parabola with vertex at 1h, k2. diameter of 20 feet. If the receiver is to be placed 6 feet from the base, how deep should the dish be? 89. Find the focus and directrix of a parabola whose equation is Writing in Mathematics of the form Ax2 + Ey = 0, A Z 0, E Z 0. 90. Write the standard form of the equation of a parabola whose points are equidistant from y = 4 and 1- 1, 02. 69. What is a parabola? 70. Explain how to use y2 = 8x to find the parabola’s focus and directrix. Group Exercise 71. If you are given the standard form of the equation of a 91. Consult the research department of your library or the parabola with vertex at the origin, explain how to determine Internet to find an example of architecture that incorporates if the parabola opens to the right, left, upward, or downward. one or more conic sections in its design. Share this example 72. Describe one similarity and one difference between the with other group members. Explain precisely how conic graphs of y2 = 4x and 1y - 122 = 41x - 12. sections are used. Do conic sections enhance the appeal of 73. How can you distinguish parabolas from other conic sections the architecture? In what ways? by looking at their equations? 74. Look at the satellite dish shown in Exercise 63. Why must the Preview Exercises receiver for a shallow dish be farther from the base of the Exercises 92–94 will help you prepare for the material covered in dish than for a deeper dish of the same diameter? the next section. 92. Simplify and write the equation in standard form in terms of Technology Exercises x¿ and y¿: 75. Use a graphing utility to graph any five of the parabolas that 22 22 you graphed by hand in Exercises 5–16. B 1x¿ - y¿2 R B 1x¿ + y¿2 R = 1. 2 2
13. P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 912 912 Chapter 9 Conic Sections and Analytic Geometry 7 93. a. Make a sketch showing that cot 2u = - for d. In part (c), why did we not write ; before the radical in 24 each formula? 90° 6 2u 6 180°. b. Use your sketch from part (a) to determine the value of 94. The equation 3x2 - 223xy + y2 + 2x + 223y = 0 is in cos 2u. the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. Use the c. Use the value of cos 2u from part (b) and the identities equation to determine the value of B2 - 4AC. 1 - cos 2u 1 + cos 2u sin u = and cos u = A 2 A 2 to determine the values of sin u and cos u. Chapter 9 Mid-Chapter Check Point What You Know: We learned that the four conic sections In Exercises 14–21, graph each equation. are the circle, the ellipse, the hyperbola, and the parabola. 14. x2 + y2 = 4 15. x + y = 4 Prior to this chapter, we graphed circles with center 1h, k2 2 2 16. x - y = 4 17. x 2 + 4y2 = 4 18. 1x + 122 + 1y - 122 = 4 and radius r: 1x - h22 + 1y - k22 = r2. 19. x 2 + 41y - 122 = 4 20. 1x - 122 - 1y - 122 = 4 21. 1y + 122 = 41x - 12 In this chapter, you learned to graph ellipses centered at the origin and ellipses centered at 1h, k2: In Exercises 22–27, find the standard form of the equation of the 1x - h22 1y - k22 conic section satisfying the given conditions. 22. Ellipse; Foci: 1-4, 02, (4, 0); Vertices: 1 - 5, 02, (5, 0) + = 1 or a2 b2 1x - h22 1y - k22 23. Ellipse; Endpoints of major axis: 1- 8, 22, (10, 2); + = 1, a2 7 b2. Foci: 1 -4, 22, (6, 2) 24. Hyperbola; Foci: 10, -32, (0, 3); Vertices: 10, - 22, (0, 2) b2 a2 We saw that the larger denominator 1a22 determines 25. Hyperbola; Foci: 1-4, 52, (2, 5); Vertices: 1- 3, 52, (1, 5) whether the major axis is horizontal or vertical. We used 26. Parabola; Focus: (4, 5); Directrix: y = - 1 27. Parabola; Focus: 1-2, 62; Directrix: x = 8 vertices and asymptotes to graph hyperbolas centered at the origin and hyperbolas centered at 1h, k2: 28. A semielliptical archway over a one-way road has a height of 1x - h22 1y - k22 1y - k22 1x - h22 10 feet and a width of 30 feet. A truck has a width of 10 feet - = 1 or - = 1. and a height of 9.5 feet. Will this truck clear the opening of a2 b2 a2 b2 the archway? We used c2 = a2 - b2 to locate the foci of an ellipse.We used 29. A lithotriper is used to disentegrate kidney stones.The patient is c2 = a2 + b2 to locate the foci of a hyperbola. Finally, we placed within an elliptical device with the kidney centered at used the vertex and the latus rectum to graph parabolas with one focus, while ultrasound waves from the other focus hit the vertices at the origin and parabolas with vertices at 1h, k2: walls and are reflected to the kidney stone, shattering the stone. 1y - k22 = 4p1x - h2 or 1x - h22 = 4p1y - k2. Suppose that the length of the major axis of the ellipse is 40 cen- timeters and the length of the minor axis is 20 centimeters. How In Exercises 1–5, graph each ellipse. Give the location of the foci. far from the kidney stone should the electrode that sends the x2 y2 ultrasound waves be placed in order to shatter the stone? 1. + = 1 2. 9x 2 + 4y2 = 36 30. An explosion is recorded by two forest rangers, one at a 25 4 1x - 222 1y + 122 1x + 222 1y - 122 primary station and the other at an outpost 6 kilometers 3. + = 1 4. + = 1 away. The ranger at the primary station hears the explosion 16 25 25 16 6 seconds before the ranger at the outpost. 5. x2 + 9y2 - 4x + 54y + 49 = 0 a. Assuming sound travels at 0.35 kilometer per second, In Exercises 6–11, graph each hyperbola. Give the location of the write an equation in standard form that gives all the foci and the equations of the asymptotes. possible locations of the explosion. Use a coordinate system with the two ranger stations on the x-axis and the x2 y2 6. - y2 = 1 7. - x2 = 1 midpoint between the stations at the origin. 9 9 b. Graph the equation that gives the possible locations of 8. y2 - 4x2 = 16 9. 4x 2 - 49y2 = 196 the explosion. Show the locations of the ranger stations 1x - 222 1y + 222 in your drawing. 10. - = 1 31. A domed ceiling is a parabolic surface. Ten meters down 9 16 from the top of the dome, the ceiling is 15 meters wide. For 11. 4x2 - y2 + 8x + 6y + 11 = 0 the best lighting on the floor, a light source should be placed In Exercises 12–13, graph each parabola. Give the location of the at the focus of the parabolic surface. How far from the top of focus and the directrix. the dome, to the nearest tenth of a meter, should the light 12. 1x - 222 = - 121y + 12 13. y2 - 2x - 2y - 5 = 0 source be placed?
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