10.2 INTRODUCTION TO CONICS: PARABOLAS

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1 Section 0.2 Introduction to Conics: Parabolas INTRODUCTION TO CONICS: PARABOLAS What ou should learn Recognize a conic as the intersection of a plane a double-napped cone. Write equations of parabolas in stard form graph parabolas. Use the reflective propert of parabolas to solve real-life problems. Wh ou should learn it Parabolas can be used to model solve man tpes of real-life problems. For instance, in Eercise 7 on page 739, 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 300 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 plaed a prominent role in the earl development of calculus. A conic section (or simpl conic) is the intersection of a plane 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 cones, as the Greeks did, or ou could define them algebraicall, in terms of the general second-degree equation A 2 B C 2 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.2, 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 stard form of the equation of a circle h 2 k 2 r 2. Equation of circle

2 734 Chapter 0 Topics in Analtic Geometr Parabolas In Section 2., ou learned that the graph of the quadratic function f a 2 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) a fied point (focus) not on the line. Directri FIGURE 0. Verte Parabola d 2 d d d 2 The midpoint between the focus the directri is called the verte, the line passing through the focus 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 stard form of the equation of a parabola whose directri is parallel to the -ais or to the -ais. Stard Equation of a Parabola The stard form of the equation of a parabola with verte at h, k is as follows. h 2 4p k, p 0 k 2 4p 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. 2 4p 2 4p See Figure 0.2. Vertical ais Horizontal ais For a proof of the stard form of the equation of a parabola, see Proofs in Mathematics on page 805. p > 0 Verte: ( h, k) Ais: = h : ( hk, + p) Directri: = k p p < 0 Ais: = h Directri: = k p Verte: (h, k) : (h, k + p) Directri: = h p p > 0 Verte: ( h, k) : ( h + p, k) Ais: =k= : (h + p, k) Directri: = h p p < 0 Verte: (h, k) Ais: = k (a) h 2 4p k Vertical ais: p > 0 FIGURE 0.2 (b) h 2 4p k Vertical ais: p < 0 (c) k 2 4p h Horizontal ais: p > 0 (d) k 2 4p h Horizontal ais: p < 0

3 Section 0.2 Introduction to Conics: Parabolas 735 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: Upper part Lower part Eample Verte at the Origin Find the stard equation of the parabola with verte at the origin focus 2, 0. The ais of the parabola is horizontal, passing through 0, 0 2, 0, as shown in Figure Verte 2 = 8 (2, 0) (0, 0) FIGURE 0.3 The stard form is 2 4p, where h 0, k 0, p 2. So, the equation is 2 8. Now tr Eercise 23. Eample 2 Finding the of a Parabola The technique of completing the square is used to write the equation in Eample 2 in stard form. You can review completing the square in Appendi A.5. 2 Verte (, ) (, 2) 3 = FIGURE 0.4 Find the focus of the parabola given b To find the focus, convert to stard form b completing the square Write original equation Multipl each side b Add to each side Complete the square Combine like terms. 2 2 Stard form Comparing this equation with h 2 4p k ou can conclude that h, k, p 2. Because p is negative, the parabola opens downward, as shown in Figure 0.4. So, the focus of the parabola is h, k p, 2. Now tr Eercise 43.

4 736 Chapter 0 Topics in Analtic Geometr Eample 3 Finding the Stard Equation of a Parabola FIGURE 0.5 Light source at focus FIGURE 0.6 ( 2) 2 = 2( ) (2, 4) Verte (2, ) Ais Parabolic reflector: Light is reflected in parallel ras. Ais P Find the stard form of the equation of the parabola with verte 2, focus 2, 4. Then write the quadratic form of the equation. Because the ais of the parabola is vertical, passing through 2, consider the equation h 2 4p k where h 2, k, p 4 3. So, the stard form is You can obtain the more common quadratic form as follows Write original equation. Multipl. Add 2 to each side. The graph of this parabola is shown in Figure 0.5. Application Now tr Eercise 55. Divide each side b 2. A line segment that passes through the focus of a parabola 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. 2, 4, FIGURE 0.7 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 the focus 2. The ais of the parabola

5 Section 0.2 Introduction to Conics: Parabolas 737 = 2 0, 4 d FIGURE 0.8 ( ) d2 (0, b) TECHNOLOGY (, ) Use a graphing utilit to confirm the result of Eample 4. B graphing in the same viewing window, ou should be able to see that the line touches the parabola at the point,. Eample 4 Finding the Tangent Line at a Point on a Parabola Find the equation of the tangent line to the parabola given b 2 at the point,. For this parabola, p the focus is 0, 4 4, 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: d 4 b d Note that d rather than b 4 b 4. The order of subtraction for the distance is important because the distance must be positive. Setting d d 2 produces 4 b 5 4 So, the slope of the tangent line is m b. 0 the equation of the tangent line in slope-intercept form is 2. 2 Now tr Eercise 65. You can review techniques for writing linear equations in Section.3. 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