Inclination of a Line. Definition of Inclination

Transcription

1 76 Chapter 0 Topics in Analtic Geometr 0. LINES What ou should learn Find the inclination of a line. Find the angle between two lines. Find the distance between a point and a line. Wh ou should learn it The inclination of a line can be used to measure heights indirectl. For instance, in Eercise 70 on page 7, the inclination of a line can be used to determine the change in elevation from the base to the top of the Falls Incline Railwa in Niagara Falls, Ontario, Canada. Inclination of a Line In Section., ou learned that the graph of the linear equation m b is a nonvertical line with slope m and -intercept 0, b. There, the slope of a line was described as the rate of change in with respect to. In this section, ou will look at the slope of a line in terms of the angle of inclination of the line. Ever nonhorizontal line must intersect the -ais. The angle formed b such an intersection determines the inclination of the line, as specified in the following definition. Definition of Inclination The inclination of a nonhorizontal line is the positive angle (less than ) measured counterclockwise from the -ais to the line. (See Figure 0..) θ = 0 JTB Photo/Japan Travel Bureau/PhotoLibrar θ = π Horizontal Line Vertical Line Acute Angle Obtuse Angle FIGURE 0. The inclination of a line is related to its slope in the following manner. θ θ Inclination and Slope If a nonvertical line has inclination and slope m, then m tan. For a proof of this relation between inclination and slope, see Proofs in Mathematics on page 80.

2 Section 0. Lines 77 Eample Finding the Inclination of a Line FIGURE 0. = θ = 5 Find the inclination of the line. The slope of this line is m. So, its inclination is determined from the equation tan. From Figure 0., it follows that 0 < arctan. The angle of inclination is radian or 5. Now tr Eercise 7. < This means that. Eample Finding the Inclination of a Line Find the inclination of the line 6. The slope of this line is m. So, its inclination is determined from the equation FIGURE 0. + = 6 θ 6. θ = θ θ tan. From Figure 0., it follows that < arctan <. This means that The angle of inclination is about.55 radians or about 6.. Now tr Eercise. FIGURE 0. θ θ θ The Angle Between Two Lines Two distinct lines in a plane are either parallel or intersecting. If the intersect and are nonperpendicular, their intersection forms two pairs of opposite angles. One pair is acute and the other pair is obtuse. The smaller of these angles is called the angle between the two lines. As shown in Figure 0., ou can use the inclinations of the two lines to find the angle between the two lines. If two lines have inclinations and, where < and <, the angle between the two lines is.

3 78 Chapter 0 Topics in Analtic Geometr You can use the formula for the tangent of the difference of two angles tan tan tan tan tan tan to obtain the formula for the angle between two lines. Angle Between Two Lines If two nonperpendicular lines have slopes m and m, the angle between the two lines is tan m m m m. Eample Finding the Angle Between Two Lines FIGURE = 0 θ = 0 Find the angle between the two lines. Line : 0 Line : 0 The two lines have slopes of m and m, respectivel. So, the tangent of the angle between the two lines is tan m m m. m Finall, ou can conclude that the angle is arctan as shown in Figure radians Now tr Eercise. The Distance Between a Point and a Line Finding the distance between a line and a point not on the line is an application of perpendicular lines. This distance is defined as the length of the perpendicular line segment joining the point and the line, as shown in Figure 0.6. (, ) (, ) d Distance Between a Point and a Line The distance between the point, and the line A B C 0 is d A B C A B. FIGURE 0.6 Remember that the values of A, B, and C in this distance formula correspond to the general equation of a line, A B C 0. For a proof of this formula for the distance between a point and a line, see Proofs in Mathematics on page 80.

4 Section 0. Lines 79 = + (, ) 5 FIGURE 0.7 Eample Finding the Distance Between a Point and a Line Find the distance between the point, and the line. The general form of the equation is 0. So, the distance between the point and the line is d 8.58 units. 5 The line and the point are shown in Figure 0.7. Now tr Eercise 5. Eample 5 An Application of Two Distance Formulas 6 5 B (0, ) h C (5, ) A (, 0) 5 FIGURE 0.8 Figure 0.8 shows a triangle with vertices A, 0, B0,, and C5,. a. Find the altitude h from verte B to side AC. b. Find the area of the triangle. a. To find the altitude, use the formula for the distance between line AC and the point 0,. The equation of line AC is obtained as follows. Slope: Equation: m Point-slope form Multipl each side b. General form So, the distance between this line and the point 0, is 0 Altitude h 7 units. b. Using the formula for the distance between two points, ou can find the length of the base AC to be b units. 0 0 Finall, the area of the triangle in Figure 0.8 is Distance Formula Simplif. Simplif. A bh 7 7 square units. Now tr Eercise 59. Formula for the area of a triangle Substitute for b and h. Simplif.

5 70 Chapter 0 Topics in Analtic Geometr 0. EXERCISES VOCABULARY: Fill in the blanks. See for worked-out solutions to odd-numbered eercises.. The of a nonhorizontal line is the positive angle (less than ) measured counterclockwise from the -ais to the line.. If a nonvertical line has inclination and slope m, then m.. If two nonperpendicular lines have slopes m and m, the angle between the two lines is tan.. The distance between the point, and the line A B C 0 is given b d. SKILLS AND APPLICATIONS In Eercises 5, find the slope of the line with inclination radians 0. radians 6. radians. radians In Eercises 8, find the inclination degrees) of the line with a slope of m m m m m m m 5 θ = π 6 θ = π θ = θ = π π (in radians and In Eercises 7 6, find the inclination degrees) of the line ,, 0,,, 6, 6,, 0, 8, 8,,, 0, 0, 0 0, 00, 50, In Eercises 7 6, find the angle between the lines. θ (in radians and (in radians and degrees) θ In Eercises 9 6, find the inclination (in radians and degrees) of the line passing through the points ,, 0,,, 0,

6 Section 0. Lines θ θ Point 5., 5., 55. 6, 56., 57. 0, 8 58., Line In Eercises 59 6, the points represent the vertices of a triangle. (a) Draw triangle ABC in the coordinate plane, (b) find the altitude from verte B of the triangle to side AC, and (c) find the area of the triangle A 0, 0, B,, C, 0 A 0, 0, B, 5, C 5, A,, B,, A, 5, B, 0, C 6, In Eercises 6 and 6, find the distance between the parallel lines C 5, 0 0 ANGLE MEASUREMENT In Eercises 7 50, find the slope of each side of the triangle and use the slopes to find the measures of the interior angles (, 8) (, 5) (, 5) (, ) (, ) (6, ) ROAD GRADE A straight road rises with an inclination of 0.0 radian from the horizontal (see figure). Find the slope of the road and the change in elevation over a two-mile stretch of the road (, ) (, 0) (, ) (, ) (, ) (, ) mi 0. radian In Eercises 5 58, find the distance between the point and the line. Point 5. 0, , 0 Line ROAD GRADE A straight road rises with an inclination of 0.0 radian from the horizontal. Find the slope of the road and the change in elevation over a one-mile stretch of the road.

7 7 Chapter 0 Topics in Analtic Geometr 67. PITCH OF A ROOF A roof has a rise of feet for ever horizontal change of 5 feet (see figure). Find the inclination of the roof. 68. CONVEYOR DESIGN A moving conveor is built so that it rises meter for each meters of horizontal travel. (a) Draw a diagram that gives a visual representation of the problem. (b) Find the inclination of the conveor. (c) The conveor runs between two floors in a factor. The distance between the floors is 5 meters. Find the length of the conveor. 69. TRUSS Find the angles and shown in the drawing of the roof truss. 70. The Falls Incline Railwa in Niagara Falls, Ontario, Canada is an inclined railwa that was designed to carr people from the Cit of Niagara Falls to Queen Victoria Park. The railwa is approimatel 70 feet long with a 6% uphill grade (see figure). θ 5 ft ft 6 ft 70 ft α β 9 ft Not drawn to scale 6 ft 6 ft (c) Using the origin of a rectangular coordinate sstem as the base of the inclined plane, find the equation of the line that models the railwa track. (d) Sketch a graph of the equation ou found in part (c). EXPLORATION TRUE OR FALSE? In Eercises 7 and 7, determine whether the statement is true or false. Justif our answer. 7. A line that has an inclination greater than radians has a negative slope. 7. To find the angle between two lines whose angles of inclination and are known, substitute and for m and m, respectivel, in the formula for the angle between two lines. 7. Consider a line with slope m and -intercept 0,. (a) Write the distance d between the origin and the line as a function of m. (b) Graph the function in part (a). (c) Find the slope that ields the maimum distance between the origin and the line. (d) Find the asmptote of the graph in part (b) and interpret its meaning in the contet of the problem. 7. CAPSTONE Discuss wh the inclination of a line can be an angle that is larger than, but the angle between two lines cannot be larger than. Decide whether the following statement is true or false: The inclination of a line is the angle between the line and the -ais. Eplain. 75. Consider a line with slope m and -intercept 0,. (a) Write the distance d between the point, and the line as a function of m. (b) Graph the function in part (a). (c) Find the slope that ields the maimum distance between the point and the line. (d) Is it possible for the distance to be 0? If so, what is the slope of the line that ields a distance of 0? (e) Find the asmptote of the graph in part (b) and interpret its meaning in the contet of the problem. (a) Find the inclination of the railwa. (b) Find the change in elevation from the base to the top of the railwa.

8 Section 0. Introduction to Conics: Parabolas 7 0. INTRODUCTION TO CONICS: PARABOLAS What ou should learn Recognize a conic as the intersection of a plane and a double-napped cone. Write equations of parabolas in standard form and graph parabolas. Use the reflective propert of parabolas to solve real-life problems. Wh ou should learn it Parabolas can be used to model and solve man tpes of real-life problems. For instance, in Eercise 7 on page 79, a parabola is used to model the cables of the Golden Gate Bridge. Conics Conic sections were discovered during the classical Greek period, 600 to 00 B.C. The earl Greeks were concerned largel with the geometric properties of conics. It was not until the 7th centur that the broad applicabilit of conics became apparent and plaed a prominent role in the earl development of calculus. A conic section (or simpl conic) is the intersection of a plane and a doublenapped cone. Notice in Figure 0.9 that in the formation of the four basic conics, the intersecting plane does not pass through the verte of the cone. When the plane does pass through the verte, the resulting figure is a degenerate conic, as shown in Figure 0.0. Cosmo Condina/The Image Bank/ Gett Images Circle Ellipse Parabola Hperbola FIGURE 0.9 Basic Conics Point Line Two Intersecting FIGURE 0.0 Degenerate Conics Lines There are several was to approach the stud of conics. You could begin b defining conics in terms of the intersections of planes and cones, as the Greeks did, or ou could define them algebraicall, in terms of the general second-degree equation A B C D E F 0. However, ou will stud a third approach, in which each of the conics is defined as a locus (collection) of points satisfing a geometric propert. For eample, in Section., ou learned that a circle is defined as the collection of all points, that are equidistant from a fied point h, k. This leads to the standard form of the equation of a circle h k r. Equation of circle

9 7 Chapter 0 Topics in Analtic Geometr Parabolas In Section., ou learned that the graph of the quadratic function f a b c is a parabola that opens upward or downward. The following definition of a parabola is more general in the sense that it is independent of the orientation of the parabola. Definition of Parabola A parabola is the set of all points, in a plane that are equidistant from a fied line (directri) and a fied point (focus) not on the line. Directri FIGURE 0. Focus Verte Parabola d d d d The midpoint between the focus and the directri is called the verte, and the line passing through the focus and the verte is called the ais of the parabola. Note in Figure 0. that a parabola is smmetric with respect to its ais. Using the definition of a parabola, ou can derive the following standard form of the equation of a parabola whose directri is parallel to the -ais or to the -ais. Standard Equation of a Parabola The standard form of the equation of a parabola with verte at h, k is as follows. h p k, p 0 k p h, p 0 Vertical ais, directri: k p Horizontal ais, directri: h p The focus lies on the ais p units (directed distance) from the verte. If the verte is at the origin 0, 0, the equation takes one of the following forms. p p See Figure 0.. Vertical ais Horizontal ais For a proof of the standard form of the equation of a parabola, see Proofs in Mathematics on page 805. p > 0 Verte: ( h, k) Ais: = h Focus: ( h, k + p) Directri: = k p p < 0 Ais: = h Directri: = k p Verte: (h, k) Focus: (h, k + p) Directri: = h p p > 0 Verte: ( h, k) Focus: ( h + p, k) Ais: = k Focus: (h + p, k) Directri: = h p p < 0 Verte: (h, k) Ais: = k (a) h p k Vertical ais: p > 0 FIGURE 0. (b) h p k Vertical ais: p < 0 (c) k p h Horizontal ais: p > 0 (d) k p h Horizontal ais: p < 0

10 Section 0. Introduction to Conics: Parabolas 75 TECHNOLOGY Use a graphing utilit to confirm the equation found in Eample. In order to graph the equation, ou ma have to use two separate equations: and 8 8. Upper part Lower part Eample Verte at the Origin Find the standard equation of the parabola with verte at the origin and focus, 0. The ais of the parabola is horizontal, passing through 0, 0 and, 0, as shown in Figure 0.. Verte = 8 Focus (, 0) (0, 0) FIGURE 0. The standard form is p, where h 0, k 0, and p. So, the equation is 8. Now tr Eercise. Eample Finding the Focus of a Parabola The technique of completing the square is used to write the equation in Eample in standard form. You can review completing the square in Appendi A.5. Verte (, ) Focus (, ) = + FIGURE 0. Find the focus of the parabola given b. To find the focus, convert to standard form b completing the square. Write original equation. Multipl each side b. Add to each side. Complete the square. Combine like terms. Standard form Comparing this equation with h p k ou can conclude that h, k, and p. Because p is negative, the parabola opens downward, as shown in Figure 0.. So, the focus of the parabola is h, k p,. Now tr Eercise.

11 76 Chapter 0 Topics in Analtic Geometr Eample Finding the Standard Equation of a Parabola 8 6 FIGURE 0.5 Light source at focus FIGURE 0.6 Focus Focus ( ) = ( ) Focus (, ) Verte (, ) 6 8 Ais Parabolic reflector: Light is reflected in parallel ras. α Ais P Find the standard form of the equation of the parabola with verte, and focus,. Then write the quadratic form of the equation. Because the ais of the parabola is vertical, passing through, consider the equation h p k where h, k, and p. So, the standard form is. You can obtain the more common quadratic form as follows. 6 Write original equation. Multipl. Add to each side. The graph of this parabola is shown in Figure 0.5. Application 6 Now tr Eercise 55. Divide each side b. A line segment that passes through the focus of a parabola and has endpoints on the parabola is called a focal chord. The specific focal chord perpendicular to the ais of the parabola is called the latus rectum. Parabolas occur in a wide variet of applications. For instance, a parabolic reflector can be formed b revolving a parabola around its ais. The resulting surface has the propert that all incoming ras parallel to the ais are reflected through the focus of the parabola. This is the principle behind the construction of the parabolic mirrors used in reflecting telescopes. Conversel, the light ras emanating from the focus of a parabolic reflector used in a flashlight are all parallel to one another, as shown in Figure 0.6. A line is tangent to a parabola at a point on the parabola if the line intersects, but does not cross, the parabola at the point. Tangent lines to parabolas have special properties related to the use of parabolas in constructing reflective surfaces. and,, FIGURE 0.7 α Tangent line Reflective Propert of a Parabola The tangent line to a parabola at a point P makes equal angles with the following two lines (see Figure 0.7).. The line passing through P and the focus. The ais of the parabola

12 Section 0. Introduction to Conics: Parabolas 77 = 0, d α FIGURE 0.8 ( ) d α (0, b) TECHNOLOGY (, ) Use a graphing utilit to confirm the result of Eample. B graphing and in the same viewing window, ou should be able to see that the line touches the parabola at the point,. Eample Finding the Tangent Line at a Point on a Parabola Find the equation of the tangent line to the parabola given b at the point,. For this parabola, p and the focus is 0,, as shown in Figure 0.8. You can find the -intercept 0, b of the tangent line b equating the lengths of the two sides of the isosceles triangle shown in Figure 0.8: and d b d 0 5. Note that d rather than b b. The order of subtraction for the distance is important because the distance must be positive. Setting d d produces b 5 So, the slope of the tangent line is m b. 0 and the equation of the tangent line in slope-intercept form is. Now tr Eercise 65. You can review techniques for writing linear equations in Section.. CLASSROOM DISCUSSION Satellite Dishes Cross sections of satellite dishes are parabolic in shape. Use the figure shown to write a paragraph eplaining wh satellite dishes are parabolic. Amplifier Dish reflector Cable to radio or TV

13 78 Chapter 0 Topics in Analtic Geometr 0. EXERCISES VOCABULARY: Fill in the blanks.. A is the intersection of a plane and a double-napped cone. See for worked-out solutions to odd-numbered eercises.. When a plane passes through the verte of a double-napped cone, the intersection is a.. A collection of points satisfing a geometric propert can also be referred to as a of points.. A is defined as the set of all points, in a plane that are equidistant from a fied line, called the, and a fied point, called the, not on the line. 5. The line that passes through the focus and the verte of a parabola is called the of the parabola. 6. The of a parabola is the midpoint between the focus and the directri. 7. A line segment that passes through the focus of a parabola and has endpoints on the parabola is called a. 8. A line is to a parabola at a point on the parabola if the line intersects, but does not cross, the parabola at the point. SKILLS AND APPLICATIONS In Eercises 9, describe in words how a plane could intersect with the double-napped cone shown to form the conic section. (e) (f) 6 9. Circle 0. Ellipse. Parabola. Hperbola In Eercises 8, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (c) (b) (d) In Eercises 9, find the standard form of the equation of the parabola with the given characteristic(s) and verte at the origin (, 6) 8 (, 6). Focus: 0,. Focus:, 0. Focus:, 0. Focus: 0, 5. Directri: 6. Directri: 7. Directri: 8. Directri: 9. Vertical ais and passes through the point, 6 0. Vertical ais and passes through the point,. Horizontal ais and passes through the point, 5. Horizontal ais and passes through the point, 8 8

14 Section 0. Introduction to Conics: Parabolas 79 In Eercises 6, find the verte, focus, and directri of the parabola, and sketch its graph In Eercises 7 50, find the verte, focus, and directri of the parabola. Use a graphing utilit to graph the parabola In Eercises 5 60, find the standard form of the equation of the parabola with the given characteristics (, 0) (, ) (, 0) 8 6 (0, ) Verte:, ; focus: 6, 56. Verte:, ; focus:, Verte: 0, ; directri: 58. Verte:, ; directri: 59. Focus:, ; directri: 60. Focus: 0, 0; directri: 8 8 (, ) In Eercises 6 and 6, change the equation of the parabola so that its graph matches the description ; upper half of parabola 6. ; lower half of parabola 8 (0, 0) (.5, ) (5, ) In Eercises 6 and 6, the equations of a parabola and a tangent line to the parabola are given. Use a graphing utilit to graph both equations in the same viewing window. Determine the coordinates of the point of tangenc Parabola Tangent Line In Eercises 65 68, find an equation of the tangent line to the parabola at the given point, and find the -intercept of the line. 65.,, 8 66.,, 9 67.,, 68.,, REVENUE The revenue R (in dollars) generated b the sale of units of a patio furniture set is given b 06 R,05. 5 Use a graphing utilit to graph the function and approimate the number of sales that will maimize revenue. 70. REVENUE The revenue R (in dollars) generated b the sale of units of a digital camera is given b 5 5 R 5, Use a graphing utilit to graph the function and approimate the number of sales that will maimize revenue. 7. SUSPENSION BRIDGE Each cable of the Golden Gate Bridge is suspended (in the shape of a parabola) between two towers that are 80 meters apart. The top of each tower is 5 meters above the roadwa. The cables touch the roadwa midwa between the towers. (a) Draw a sketch of the bridge. Locate the origin of a rectangular coordinate sstem at the center of the roadwa. Label the coordinates of the known points. (b) Write an equation that models the cables. (c) Complete the table b finding the height of the suspension cables over the roadwa at a distance of meters from the center of the bridge. Distance, Height,

15 70 Chapter 0 Topics in Analtic Geometr 7. SATELLITE DISH The receiver in a parabolic satellite dish is.5 feet from the verte and is located at the focus (see figure). Write an equation for a cross section of the reflector. (Assume that the dish is directed upward and the verte is at the origin.) Receiver.5 ft 75. BEAM DEFLECTION A simpl supported beam is meters long and has a load at the center (see figure). The deflection of the beam at its center is centimeters. Assume that the shape of the deflected beam is parabolic. (a) Write an equation of the parabola. (Assume that the origin is at the center of the deflected beam.) (b) How far from the center of the beam is the deflection equal to centimeter? cm 7. ROAD DESIGN Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road that is feet wide is 0. foot higher in the center than it is on the sides (see figure). Cross section of road surface (a) Find an equation of the parabola that models the road surface. (Assume that the origin is at the center of the road.) (b) How far from the center of the road is the road surface 0. foot lower than in the middle? 7. HIGHWAY DESIGN Highwa engineers design a parabolic curve for an entrance ramp from a straight street to an interstate highwa (see figure). Find an equation of the parabola Interstate ft (000, 800) ft Not drawn to scale 76. BEAM DEFLECTION Repeat Eercise 75 if the length of the beam is 6 meters and the deflection of the beam at the center is centimeters. 77. FLUID FLOW Water is flowing from a horizontal pipe 8 feet above the ground. The falling stream of water has the shape of a parabola whose verte 0, 8 is at the end of the pipe (see figure). The stream of water strikes the ground at the point 0, 0. Find the equation of the path taken b the water ft FIGURE FOR 77 FIGURE FOR 78 m 6 (0, 6) (, 6) (, 6) LATTICE ARCH A parabolic lattice arch is 6 feet high at the verte. At a height of 6 feet, the width of the lattice arch is feet (see figure). How wide is the lattice arch at ground level? 79. SATELLITE ORBIT A satellite in a 00-mile-high circular orbit around Earth has a velocit of approimatel 7,500 miles per hour. If this velocit is multiplied b, the satellite will have the minimum velocit necessar to escape Earth s gravit and it will follow a parabolic path with the center of Earth as the focus (see figure on the net page). Not drawn to scale 800 (000, 800) Street

16 Section 0. Introduction to Conics: Parabolas 7 FIGURE FOR 79 (a) Find the escape velocit of the satellite. (b) Find an equation of the parabolic path of the satellite (assume that the radius of Earth is 000 miles). 80. PATH OF A SOFTBALL The path of a softball is modeled b , where the coordinates and are measured in feet, with 0 corresponding to the position from which the ball was thrown. (a) Use a graphing utilit to graph the trajector of the softball. (b) Use the trace feature of the graphing utilit to approimate the highest point and the range of the trajector. PROJECTILE MOTION In Eercises 8 and 8, consider the path of a projectile projected horizontall with a velocit of v feet per second at a height of s feet, where the model for the path is In this model (in which air resistance is disregarded), is the height (in feet) of the projectile and is the horizontal distance (in feet) the projectile travels. 8. A ball is thrown from the top of a 00-foot tower with a velocit of 8 feet per second. (a) Find the equation of the parabolic path. (b) How far does the ball travel horizontall before striking the ground? 8. A cargo plane is fling at an altitude of 0,000 feet and a speed of 50 miles per hour. A suppl crate is dropped from the plane. How man feet will the crate travel horizontall before it hits the ground? EXPLORATION 00 miles v s. 6 Circular orbit TRUE OR FALSE? In Eercises 8 and 8, determine whether the statement is true or false. Justif our answer. 8. It is possible for a parabola to intersect its directri. 8. If the verte and focus of a parabola are on a horizontal line, then the directri of the parabola is vertical. Parabolic path Not drawn to scale 85. Let, be the coordinates of a point on the parabola p. The equation of the line tangent to the parabola at the point is p. What is the slope of the tangent line? 86. CAPSTONE Eplain what each of the following equations represents, and how equations (a) and (b) are equivalent. (a) a h k, a 0 (b) h p k, p 0 (c) k p h, p GRAPHICAL REASONING Consider the parabola p. (a) Use a graphing utilit to graph the parabola for p, p, p, and p. Describe the effect on the graph when p increases. (b) Locate the focus for each parabola in part (a). (c) For each parabola in part (a), find the length of the latus rectum (see figure). How can the length of the latus rectum be determined directl from the standard form of the equation of the parabola? Latus rectum (d) Eplain how the result of part (c) can be used as a sketching aid when graphing parabolas. 88. GEOMETRY The area of the shaded region in the figure is A 8 p b. = p (a) Find the area when p and b. Focus = p = b (b) Give a geometric eplanation of wh the area approaches 0 as p approaches 0.

17 7 Chapter 0 Topics in Analtic Geometr 0. ELLIPSES What ou should learn Write equations of ellipses in standard form and graph ellipses. Use properties of ellipses to model and solve real-life problems. Find eccentricities of ellipses. Wh ou should learn it Ellipses can be used to model and solve man tpes of real-life problems. For instance, in Eercise 65 on page 79, an ellipse is used to model the orbit of Halle s comet. Introduction The second tpe of conic is called an ellipse, and is defined as follows. Definition of Ellipse An ellipse is the set of all points, in a plane, the sum of whose distances from two distinct fied points (foci) is constant. See Figure 0.9. Focus d (, ) d Focus Major ais Verte Center Minor ais Verte Harvard College Observator/Photo Researchers, Inc. d d is constant. FIGURE 0.9 FIGURE 0.0 The line through the foci intersects the ellipse at two points called vertices. The chord joining the vertices is the major ais, and its midpoint is the center of the ellipse. The chord perpendicular to the major ais at the center is the minor ais of the ellipse. See Figure 0.0. You can visualize the definition of an ellipse b imagining two thumbtacks placed at the foci, as shown in Figure 0.. If the ends of a fied length of string are fastened to the thumbtacks and the string is drawn taut with a pencil, the path traced b the pencil will be an ellipse. (, ) FIGURE 0. b + c b + c ( h, k) b To derive the standard form of the equation of an ellipse, consider the ellipse in Figure 0. with the following points: center, h, k; vertices, h ± a, k; foci, h ± c, k. Note that the center is the midpoint of the segment joining the foci. The sum of the distances from an point on the ellipse to the two foci is constant. Using a verte point, this constant sum is b + c = a b + c = a FIGURE 0. c a a c a c a Length of major ais or simpl the length of the major ais. Now, if ou let, be an point on the ellipse, the sum of the distances between, and the two foci must also be a.

18 Section 0. Ellipses 7 That is, h c k h c k a which, after epanding and regrouping, reduces to a c h a k a a c. Finall, in Figure 0., ou can see that b a c which implies that the equation of the ellipse is b h a k a b h a k b. You would obtain a similar equation in the derivation b starting with a vertical major ais. Both results are summarized as follows. Consider the equation of the ellipse h a k b. If ou let a b, then the equation can be rewritten as h k a which is the standard form of the equation of a circle with radius r a (see Section.). Geometricall, when a b for an ellipse, the major and minor aes are of equal length, and so the graph is a circle. Standard Equation of an Ellipse The standard form of the equation of an ellipse, with center h, k and major and minor aes of lengths a and b, respectivel, where 0 < b < a, is h h Figure 0. shows both the horizontal and vertical orientations for an ellipse. a b Major ais is horizontal. Major ais is vertical. The foci lie on the major ais, c units from the center, with c a b. If the center is at the origin 0, 0, the equation takes one of the following forms. a b k b k a. Major ais is horizontal. b a Major ais is vertical. ( h) ( k) + = a b ( h) ( k) + = b a (h, k) b (h, k) a a b Major ais is horizontal. FIGURE 0. Major ais is vertical.

19 7 Chapter 0 Topics in Analtic Geometr Eample Finding the Standard Equation of an Ellipse FIGURE 0. (0, ) (, ) (, ) a = b = 5 Find the standard form of the equation of the ellipse having foci at 0, and, and a major ais of length 6, as shown in Figure 0.. Because the foci occur at 0, and,, the center of the ellipse is, ) and the distance from the center to one of the foci is c. Because a 6, ou know that a. Now, from c a b, ou have b a c 5. Because the major ais is horizontal, the standard equation is. 5 This equation simplifies to 9 5. Now tr Eercise. Eample Sketching an Ellipse ( + ) ( ) + = ( 5, ) (, ) (, ) (, ) (, ) ( +, ) 5 (, 0) FIGURE 0.5 Sketch the ellipse given b Begin b writing the original equation in standard form. In the fourth step, note that 9 and are added to both sides of the equation when completing the squares Write original equation. Group terms Factor out of -terms. Write in completed square form. Divide each side b. Write in standard form. From this standard form, it follows that the center is h, k,. Because the denominator of the -term is a, the endpoints of the major ais lie two units to the right and left of the center. Similarl, because the denominator of the -term is b, the endpoints of the minor ais lie one unit up and down from the center. Now, from c a b, ou have c. So, the foci of the ellipse are, and,. The ellipse is shown in Figure 0.5. Now tr Eercise 7.

20 Section 0. Ellipses 75 Eample Analzing an Ellipse ( ) ( + ) + = Verte, + Focus ( ( (, ), FIGURE 0.6 ( ( Center Focus Verte (, ) (, 6) Find the center, vertices, and foci of the ellipse B completing the square, ou can write the original equation in standard form Write original equation. Group terms. The major ais is vertical, where h, k, a, b, and c a b 6. So, ou have the following Center:, Vertices:, 6 Foci: The graph of the ellipse is shown in Figure 0.6. Now tr Eercise 5. 6, Factor out of -terms. Write in completed square form. Divide each side b 6. Write in standard form.,, TECHNOLOGY You can use a graphing utilit to graph an ellipse b graphing the upper and lower portions in the same viewing window. For instance, to graph the ellipse in Eample, first solve for to get and Use a viewing window in which 6 9 and 7. You should obtain the graph shown below

21 76 Chapter 0 Topics in Analtic Geometr Application Ellipses have man practical and aesthetic uses. For instance, machine gears, supporting arches, and acoustic designs often involve elliptical shapes. The orbits of satellites and planets are also ellipses. Eample investigates the elliptical orbit of the moon about Earth. Eample An Application Involving an Elliptical Orbit 767,60 km Moon The moon travels about Earth in an elliptical orbit with Earth at one focus, as shown in Figure 0.7. The major and minor aes of the orbit have lengths of 768,800 kilometers and 767,60 kilometers, respectivel. Find the greatest and smallest distances (the apogee and perigee, respectivel) from Earth s center to the moon s center. Earth Perigee FIGURE ,800 km Apogee Because a 768,800 and b 767,60, ou have a 8,00 and b 8,80 which implies that c a b 8,00 8,80,08. So, the greatest distance between the center of Earth and the center of the moon is WARNING / CAUTION Note in Eample and Figure 0.7 that Earth is not the center of the moon s orbit. a c 8,00,08 05,508 kilometers and the smallest distance is a c 8,00,08 6,9 kilometers. Now tr Eercise 65. Eccentricit One of the reasons it was difficult for earl astronomers to detect that the orbits of the planets are ellipses is that the foci of the planetar orbits are relativel close to their centers, and so the orbits are nearl circular. To measure the ovalness of an ellipse, ou can use the concept of eccentricit. Definition of Eccentricit The eccentricit e of an ellipse is given b the ratio e c a. Note that 0 < e < for ever ellipse.

22 Section 0. Ellipses 77 To see how this ratio is used to describe the shape of an ellipse, note that because the foci of an ellipse are located along the major ais between the vertices and the center, it follows that 0 < c < a. For an ellipse that is nearl circular, the foci are close to the center and the ratio c a is small, as shown in Figure 0.8. On the other hand, for an elongated ellipse, the foci are close to the vertices and the ratio c a is close to, as shown in Figure 0.9. Foci Foci c e= a c e= c a e is small. c e is close to. a FIGURE 0.8 a FIGURE 0.9 The orbit of the moon has an eccentricit of e " 0.059, and the eccentricities of the eight planetar orbits are as follows. e " e " Jupiter: Saturn: Uranus: Neptune: e " e " 0.09 e " 0.08 e " 0.05 e " 0.07 e " NASA Mercur: Venus: Earth: Mars: The time it takes Saturn to orbit the sun is about 9. Earth ears. CLASSROOM DISCUSSION Ellipses and Circles a. Show that the equation of an ellipse can be written as h! a " a k!!. e! b. For the equation in part (a), let a!, h!, and k!, and use a graphing utilit to graph the ellipse for e! 0.95, e! 0.75, e! 0.5, e! 0.5, and e! 0.. Discuss the changes in the shape of the ellipse as e approaches 0. c. Make a conjecture about the shape of the graph in part (b) when e! 0. What is the equation of this ellipse? What is another name for an ellipse with an eccentricit of 0?

23 78 Chapter 0 Topics in Analtic Geometr 0. EXERCISES VOCABULARY: Fill in the blanks. See for worked-out solutions to odd-numbered eercises.. An is the set of all points, in a plane, the sum of whose distances from two distinct fied points, called, is constant.. The chord joining the vertices of an ellipse is called the, and its midpoint is the of the ellipse.. The chord perpendicular to the major ais at the center of the ellipse is called the of the ellipse.. The concept of is used to measure the ovalness of an ellipse. SKILLS AND APPLICATIONS In Eercises 5 0, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (c) (e) (b) (d) (f) In Eercises 8, find the standard form of the equation of the ellipse with the given characteristics and center at the origin Vertices: ±7, 0; foci: ±, 0. Vertices: 0, ±8; foci: 0, ± 5. Foci: ±5, 0; major ais of length 6. Foci: ±, 0; major ais of length 0 7. Vertices: 0, ±5; passes through the point, 8. Vertical major ais; passes through the points 0, 6 and, 0 In Eercises 9 8, find the standard form of the equation of the ellipse with the given characteristics (, ) (, 6) (0, ) (, 0) (, 0) 8 (0, ) (, ) (, 0) 5 6 (, 0) (0, ). Vertices: 0,, 8, ; minor ais of length. Foci: 0, 0,, 0; major ais of length 6 (, ) (, ). Foci: 0, 0, 0, 8; major ais of length 6. Center:, ; verte:, ; minor ais of length 5. Center: 0, ; a c; vertices:,,, 6. Center:, ; a c; foci:,, 5, ( 0, ) (, 0) (, 0) ( 0, )

24 Section 0. Ellipses Vertices: 0,,, ; endpoints of the minor ais:,,, 8. Vertices: 5, 0, 5, ; endpoints of the minor ais:, 6, 9, 6 In Eercises 9 5, identif the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricit of the conic (if applicable), and sketch its graph In Eercises 5 56, use a graphing utilit to graph the ellipse. Find the center, foci, and vertices. (Recall that it ma be necessar to solve the equation for and obtain two equations.) In Eercises 57 60, find the eccentricit of the ellipse Find an equation of the ellipse with vertices ±5, 0 and eccentricit e Find an equation of the ellipse with vertices 0, ±8 and eccentricit e. 6. ARCHITECTURE A semielliptical arch over a tunnel for a one-wa road through a mountain has a major ais of 50 feet and a height at the center of 0 feet. (a) Draw a rectangular coordinate sstem on a sketch of the tunnel with the center of the road entering the tunnel at the origin. Identif the coordinates of the known points. (b) Find an equation of the semielliptical arch. (c) You are driving a moving truck that has a width of 8 feet and a height of 9 feet. Will the moving truck clear the opening of the arch? 6. ARCHITECTURE A fireplace arch is to be constructed in the shape of a semiellipse. The opening is to have a height of feet at the center and a width of 6 feet along the base (see figure). The contractor draws the outline of the ellipse using tacks as described at the beginning of this section. Determine the required positions of the tacks and the length of the string COMET ORBIT Halle s comet has an elliptical orbit, with the sun at one focus. The eccentricit of the orbit is approimatel The length of the major ais of the orbit is approimatel 5.88 astronomical units. (An astronomical unit is about 9 million miles.) (a) Find an equation of the orbit. Place the center of the orbit at the origin, and place the major ais on the -ais. (b) Use a graphing utilit to graph the equation of the orbit. (c) Find the greatest (aphelion) and smallest (perihelion) distances from the sun s center to the comet s center.

25 750 Chapter 0 Topics in Analtic Geometr 66. SATELLITE ORBIT The first artificial satellite to orbit Earth was Sputnik I (launched b the former Soviet Union in 957). Its highest point above Earth s surface was 97 kilometers, and its lowest point was 8 kilometers (see figure). The center of Earth was at one focus of the elliptical orbit, and the radius of Earth is 678 kilometers. Find the eccentricit of the orbit. EXPLORATION TRUE OR FALSE? In Eercises 7 and 7, determine whether the statement is true or false. Justif our answer. 7. The graph of 0 is an ellipse. 7. It is easier to distinguish the graph of an ellipse from the graph of a circle if the eccentricit of the ellipse is large (close to ). 75. Consider the ellipse 67. MOTION OF A PENDULUM The relation between the velocit (in radians per second) of a pendulum and its angular displacement from the vertical can be modeled b a semiellipse. A -centimeter pendulum crests 0 when the angular displacement is 0. radian and 0. radian. When the pendulum is at equilibrium the velocit is.6 radians per second. 0, Focus 8 km 97 km (a) Find an equation that models the motion of the pendulum. Place the center at the origin. (b) Graph the equation from part (a). (c) Which half of the ellipse models the motion of the pendulum? 68. GEOMETRY A line segment through a focus of an ellipse with endpoints on the ellipse and perpendicular to the major ais is called a latus rectum of the ellipse. Therefore, an ellipse has two latera recta. Knowing the length of the latera recta is helpful in sketching an ellipse because it ields other points on the curve (see figure). Show that the length of each latus rectum is b a. F F Latera recta a, b a b 0. (a) The area of the ellipse is given b A ab. Write the area of the ellipse as a function of a. (b) Find the equation of an ellipse with an area of 6 square centimeters. (c) Complete the table using our equation from part (a), and make a conjecture about the shape of the ellipse with maimum area. a A (d) Use a graphing utilit to graph the area function and use the graph to support our conjecture in part (c). 76. THINK ABOUT IT At the beginning of this section it was noted that an ellipse can be drawn using two thumbtacks, a string of fied length (greater than the distance between the two tacks), and a pencil. If the ends of the string are fastened at the tacks and the string is drawn taut with a pencil, the path traced b the pencil is an ellipse. (a) What is the length of the string in terms of a? (b) Eplain wh the path is an ellipse. 77. THINK ABOUT IT Find the equation of an ellipse such that for an point on the ellipse, the sum of the distances from the point, and 0, is CAPSTONE Describe the relationship between circles and ellipses. How are the similar? How do the differ? In Eercises 69 7, sketch the graph of the ellipse, using latera recta (see Eercise 68) PROOF Show that a b c for the ellipse a b where a > 0, b > 0, and the distance from the center of the ellipse 0, 0 to a focus is c.

26 Section 0. Hperbolas HYPERBOLAS What ou should learn Write equations of hperbolas in standard form. Find asmptotes of and graph hperbolas. Use properties of hperbolas to solve real-life problems. Classif conics from their general equations. Wh ou should learn it Hperbolas can be used to model and solve man tpes of real-life problems. For instance, in Eercise 5 on page 759, hperbolas are used in long distance radio navigation for aircraft and ships. Introduction The third tpe of conic is called a hperbola. The definition of a hperbola is similar to that of an ellipse. The difference is that for an ellipse the sum of the distances between the foci and a point on the ellipse is fied, whereas for a hperbola the difference of the distances between the foci and a point on the hperbola is fied. Definition of Hperbola A hperbola is the set of all points, in a plane, the difference of whose distances from two distinct fied points (foci) is a positive constant. See Figure 0.0. Focus d (, ) d Focus d d is a positive constant. Branch FIGURE 0.0 FIGURE 0. Verte a Center c Transverse ais Verte Branch U.S. Nav, William Lipski/AP Photo The graph of a hperbola has two disconnected branches. The line through the two foci intersects the hperbola at its two vertices. The line segment connecting the vertices is the transverse ais, and the midpoint of the transverse ais is the center of the hperbola. See Figure 0.. The development of the standard form of the equation of a hperbola is similar to that of an ellipse. Note in the definition below that a, b, and c are related differentl for hperbolas than for ellipses. Standard Equation of a Hperbola The standard form of the equation of a hperbola with center h, k is h a k a Transverse ais is horizontal. Transverse ais is vertical. The vertices are a units from the center, and the foci are c units from the center. Moreover, c a b. If the center of the hperbola is at the origin 0, 0, the equation takes one of the following forms. a b k b h b. Transverse ais is horizontal. a b Transverse ais is vertical.

27 75 Chapter 0 Topics in Analtic Geometr Figure 0. shows both the horizontal and vertical orientations for a hperbola. ( h) ( k) = a b ( k) ( h) = a b ( h, k + c) ( h c, k) ( h, k) ( h + c, k) ( h, k) Transverse ais is horizontal. FIGURE 0. ( h, k c) Transverse ais is vertical. Eample Finding the Standard Equation of a Hperbola When finding the standard form of the equation of an conic, it is helpful to sketch a graph of the conic with the given characteristics. Find the standard form of the equation of the hperbola with foci, and 5, and vertices 0, and,. B the Midpoint Formula, the center of the hperbola occurs at the point,. Furthermore, c 5 and a, and it follows that b c a 9 5. So, the hperbola has a horizontal transverse ais and the standard form of the equation is 5. This equation simplifies to 5. See Figure 0.. ( ) ( ) = ( 5 ( 5 (0, ) (, ) (, ) (, ) (5, ) FIGURE 0. Now tr Eercise 5.

28 Section 0. Hperbolas 75 Conjugate ais (h, k + b) Asmptote Asmptotes of a Hperbola Each hperbola has two asmptotes that intersect at the center of the hperbola, as shown in Figure 0.. The asmptotes pass through the vertices of a rectangle of dimensions a b b, with its center at h, k. The line segment of length b joining h, k b and h, k b or h b, k and h b, k is the conjugate ais of the hperbola. FIGURE 0. (h, k) (h, k b) (h a, k) (h + a, k) Asmptote Asmptotes of a Hperbola The equations of the asmptotes of a hperbola are k ± b h a k ± a h. b Transverse ais is horizontal. Transverse ais is vertical. Eample Using Asmptotes to Sketch a Hperbola Sketch the hperbola whose equation is 6. Algebraic Divide each side of the original equation b 6, and rewrite the equation in standard form. Write in standard form. From this, ou can conclude that a, b, and the transverse ais is horizontal. So, the vertices occur at, 0 and, 0, and the endpoints of the conjugate ais occur at 0, and 0,. Using these four points, ou are able to sketch the rectangle shown in Figure 0.5. Now, from c a b, ou have c 0 5. So, the foci of the hperbola are 5, 0 and 5, 0. Finall, b drawing the asmptotes through the corners of this rectangle, ou can complete the sketch shown in Figure 0.6. Note that the asmptotes are and. Graphical Solve the equation of the hperbola for as follows. 6 6 ± 6 Then use a graphing utilit to graph 6 and 6 in the same viewing window. Be sure to use a square setting. From the graph in Figure 0.7, ou can see that the transverse ais is horizontal. You can use the zoom and trace features to approimate the vertices to be, 0 and, 0. 6 = 6 8 (0, ) = 6 6 (, 0) 6 (, 0) 6 (0, ) ( 5, 0) 6 6 ( 5, 0) 6 = FIGURE 0.7 FIGURE 0.5 FIGURE 0.6 Now tr Eercise.

29 75 Chapter 0 Topics in Analtic Geometr Eample Finding the Asmptotes of a Hperbola Sketch the hperbola given b and find the equations of its asmptotes and the foci Write original equation. Group terms. Factor from -terms. Add to each side. Write in completed square form. (, 7) 5 (, ) ( + ) = (, 0) ( ) 5 (, ) (, 7 ) FIGURE 0.8 Divide each side b. Write in standard form. From this equation ou can conclude that the hperbola has a vertical transverse ais, centered at, 0, has vertices, and,, and has a conjugate ais with endpoints, 0 and, 0. To sketch the hperbola, draw a rectangle through these four points. The asmptotes are the lines passing through the corners of the rectangle. Using a and b, ou can conclude that the equations of the asmptotes are and Finall, ou can determine the foci b using the equation c a b. So, ou have c 7, and the foci are,7 and, 7. The hperbola is shown in Figure 0.8. Now tr Eercise 9.. TECHNOLOGY You can use a graphing utilit to graph a hperbola b graphing the upper and lower portions in the same viewing window. For instance, to graph the hperbola in Eample, first solve for to get and. Use a viewing window in which 9 9 and 6 6. You should obtain the graph shown below. Notice that the graphing utilit does not draw the asmptotes. However, if ou trace along the branches, ou will see that the values of the hperbola approach the asmptotes

30 Section 0. Hperbolas 755 = 8 Eample Using Asmptotes to Find the Standard Equation 6 6 (, ) (, 5) Find the standard form of the equation of the hperbola having vertices, 5 and, and having asmptotes 8 and as shown in Figure 0.9. B the Midpoint Formula, the center of the hperbola is,. Furthermore, the hperbola has a vertical transverse ais with a. From the original equations, ou can determine the slopes of the asmptotes to be FIGURE 0.9 = + m and m a a b b and, because a, ou can conclude a b b b. So, the standard form of the equation is. Now tr Eercise. As with ellipses, the eccentricit of a hperbola is e c a Eccentricit and because c > a, it follows that e >. If the eccentricit is large, the branches of the hperbola are nearl flat, as shown in Figure 0.0. If the eccentricit is close to, the branches of the hperbola are more narrow, as shown in Figure 0.. e is large. e is close to. Verte Focus Verte Focus e = c a c e = c a a c a FIGURE 0.0 FIGURE 0.

31 756 Chapter 0 Topics in Analtic Geometr Applications The following application was developed during World War II. It shows how the properties of hperbolas can be used in radar and other detection sstems. Eample 5 An Application Involving Hperbolas Two microphones, mile apart, record an eplosion. Microphone A receives the sound seconds before microphone B. Where did the eplosion occur? (Assume sound travels at 00 feet per second.) 000 Assuming sound travels at 00 feet per second, ou know that the eplosion took place 00 feet farther from B than from A, as shown in Figure 0.. The locus of all points that are 00 feet closer to A than to B is one branch of the hperbola 00 B A where a b c c a c a c = ( c a) = 580 FIGURE 0. and a So, b c a ,759,600, and ou can conclude that the eplosion occurred somewhere on the right branch of the hperbola,0,000. 5,759,600 Now tr Eercise 5. Hperbolic orbit Verte Elliptical orbit Sun p Parabolic orbit FIGURE 0. Another interesting application of conic sections involves the orbits of comets in our solar sstem. Of the 60 comets identified prior to 970, 5 have elliptical orbits, 95 have parabolic orbits, and 70 have hperbolic orbits. The center of the sun is a focus of each of these orbits, and each orbit has a verte at the point where the comet is closest to the sun, as shown in Figure 0.. Undoubtedl, there have been man comets with parabolic or hperbolic orbits that were not identified. We onl get to see such comets once. Comets with elliptical orbits, such as Halle s comet, are the onl ones that remain in our solar sstem. If p is the distance between the verte and the focus (in meters), and v is the velocit of the comet at the verte (in meters per second), then the tpe of orbit is determined as follows.. Ellipse:. Parabola: v < GMp v GMp. Hperbola: v > GMp In each of these relations, M kilograms (the mass of the sun) and G cubic meter per kilogram-second squared (the universal gravitational constant).

32 Section 0. Hperbolas 757 General Equations of Conics Classifing a Conic from Its General Equation The graph of A C D E F 0 is one of the following.. Circle: A C. Parabola: AC 0 A 0 or C 0, but not both.. Ellipse: AC > 0 A and C have like signs.. Hperbola: AC < 0 A and C have unlike signs. The test above is valid if the graph is a conic. The test does not appl to equations such as, whose graph is not a conic. Eample 6 Classifing Conics from General Equations Classif the graph of each equation. a b c. 0 d. 8 0 a. For the equation 9 5 0, ou have AC 0 0. Parabola So, the graph is a parabola. b. For the equation 8 6 0, ou have HISTORICAL NOTE AC < 0. Hperbola So, the graph is a hperbola. c. For the equation 0, ou have The Granger Collection AC > 0. Caroline Herschel (750 88) was the first woman to be credited with detecting a new comet. During her long life, this English astronomer discovered a total of eight new comets. Ellipse So, the graph is an ellipse. d. For the equation 8 0, ou have A C. Circle So, the graph is a circle. Now tr Eercise 6. CLASSROOM DISCUSSION Sketching Conics Sketch each of the conics described in Eample 6. Write a paragraph describing the procedures that allow ou to sketch the conics efficientl.