10.2: Parabolas. Chapter 10: Conic Sections. Conic sections are plane figures formed by the intersection of a double-napped cone and a plane.

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1 Conic sections are plane figures formed b the intersection of a double-napped cone and a plane. Chapter 10: Conic Sections Ellipse Hperbola The conic sections ma be defined as the sets of points in the plane that satisf certain geometrical properties. 10.2: s 1

2 A parabola is the set of all points in the plane equidistant from a fied line and a fied point not on the line. The fied line is the directri. The fied point is the focus. The ais of smmetr is the line passing through the focus and perpendicular to the directri. Ais of Smmetr AoS parabola focus The verte is the midpoint of the line segment along the ais joining the directri to the focus. verte directri The standard form for the equation of a parabola with verte at the origin and a vertical ais is: vertical ais: = 0 WARNING!!! The tetbook uses a instead of p BUT we will be using a again for hperbolas and ellipses and I don t want ou to get confused with the variables so for now, we will use p 2 = 4p where p 0 directri: = p, (0, p) p focus: (0, p) = p 2 = 4p Note: p is the directed distance from the verte to the focus. The standard form for the equation of a parabola with verte at the origin and a horizontal ais is: 2 = 4p where p 0 horizontal ais: = 0, directri: = p = p p (p, 0) 2 = 4p Note: p is the directed distance from the verte to the focus. focus: ( p, 0) s with verte at V(h, k) p > 0 p < 0 s with verte at V(h, k) p > 0 p < 0 p p p p directri Equation of the Directri directri Equation of the Directri 2

3 s with verte at V(h, k) Standard equations of p > 0 opening Verte at the origin Verte at (h, k) Right 2 = 4p ( k) 2 = 4p( h) Left 2 = -4p ( k) 2 = -4p( h) Up 2 = 4p ( h) 2 = 4p( k) Down 2 = -4p ( h) 2 = -4p( k) s: Latus Rectum The latus rectum is the line segment passing through the focus, perpendicular to the ais of smmetr with endpoints on the parabola. The length of the latus rectum is 4p. Focus Verte Latus Rectum = a 2 Eample: Find the directri, focus, and verte, and sketch the parabola with equation. Rewrite the equation in standard form 2 = 4p. 2 = 8 2 = 4( 2) p = 2 verte: vertical ais: = 0 directri: = p = 2 focus: = (0, p) (0, 2) (0, 2) = 0 = 2 Eample: Write the standard form of the equation of the parabola with focus (1, 0) and directri = 1. Graph the equation.c = -1 p = 1 (1, 0) verte Use the standard from for the equation of a parabola with a horizontal ais: 2 = 4p. p = 1 2 = 4(1). The equation is 2 = 4. 3

4 Analze the given equation (i.e., find the verte, focus, and directri of the parabola then graph it.) Analze the given equation (i.e., find the verte, focus, and directri of the parabola then graph it.) Find an equation of the parabola with focus at (0, 4) and directri the line = -4. Graph the equation. Writing an Equation of a Translated Write an equation for the parabola. Writing an Equation of a Translated Write an equation for the parabola. 4

5 Writing an Equation of a Translated Write an equation for the parabola. Use a graphing calculator to graph the given equation of the parabola. Analze the given equation (i.e., find the verte, focus, and directri of the parabola then graph it.) Paraboloid of revolution a surface formed b rotating a parabola about its azis of smmetr. Paraboloid of revolution a surface formed b rotating a parabola about its azis of smmetr. 5

6 10.3: Ellipses An ellipse is the set of all points in the plane for which the sum of the distances to two fied points (called foci) is a positive constant. The major ais is the line segment passing through the foci with endpoints (called vertices) on the ellipse. The midpoint of the major ais is the center of the ellipse. verte The minor ais is the line segment perpendicular to the major ais passing through the center of the ellipse with endpoints on the ellipse. Ellipse d 1 focus d 2 focus center minor ais d 1 + d 2 = constant = 2a P major ais verte The standard form for the equation of an ellipse with center at and a major ais that is horizontal is:, with: vertices: ( a, 0), (a, 0) andfoci: ( c, 0), (c, 0) where c 2 = a 2 b 2 The standard form for the equation of an ellipse with center at (h, k) and a major ais that is horizontal is: where a > b and c 2 = a 2 b 2 ( a, 0) (0, b) a b c a ( c, 0) (c, 0) (a, 0) V 1 B 1 minor ais C(h, k) F 1 F 2 V2 major ais (0, b) B 2 6

7 The standard form for the equation of an ellipse with center at the origin and a major ais that is vertical is:, with: vertices: (0, a), (0, a) andfoci: (0, c), (0, c) where c 2 = a 2 b 2 (0, a) The standard form for the equation of an ellipse with center at (h, k) and a major ais that is vertical is: where a > b and c 2 = a 2 b 2 V 1 major ais ( b, 0) a b (0, c) c b a (0, -c) (b, 0) B 1 F 1 C(h, k) F 2 B 2 minor ais (0, a) V 2 The standard form for the equation of an ellipse with center at (h, k) Eample: Sketch the ellipse with equation = 400 where a > b and c 2 = a 2 b 2 and find the vertices and foci. 1. Put the equation into standard form. Horizontal Major Ais Vertical Major Ais divide b 400 (0, 5) major ais (0, 3) minor ais V (4, 0) B F major ais 1 1 So, a = 5 and b = 4. ( 4, 0) 4 minor ais 2. Since the denominator of the C(h, k) 2 -term F 1 C(h, k) is larger, the major ais is vertical. F 2 V2 B 1 B 2 (0, 5) (0, 3) 3. Vertices: (0, 5), (0, 5) B 2 F 2 4. The minor ais is horizontal and intersects the ellipse at ( 4, 0) and (4, 0). V 1 V 2 5. Foci: c 2 = a 2 b 2 (5) 2 (4) 2 = 9 c = 3 Foci: (0, 3), (0,3) 7

8 Analze the given equation (i.e., find the vertices, foci, and intercepts of the ellipse then graph it.) Analze the given equation (i.e., find the vertices, foci, and intercepts of the ellipse then graph it.) 10.4: Hperbolas A hperbola is the set of all points in the plane for which the difference from two fied points (the foci) is a positive constant. The graph of the hperbola has two branches. The line through the foci intersects the hperbola at two points called vertices. The line segment joining the vertices is the transverse ais. Its midpoint is the center of the hperbola. hperbola verte focus transverse ais center verte focus d 1 d2 d 2 d 1 = constant 8

9 The standard form for the equation of a hperbola with a horizontal transverse ais is: with: vertices: ( a, 0), (a, 0) andfoci: ( c, 0), (c, 0) where b 2 = c 2 a 2 verte ( a, 0) focus ( c, 0) asmptote (0, b) verte (a, 0) focus (c, 0) asmptote (0, b) A hperbola with a horizontal transverse ais has asmptotes with equations and. The standard form for the equation of a hperbola with a vertical transverse ais is: focus (0, c) with: vertices: (0, a), (0, a) andfoci: (0, c), (0, c) where b 2 = c 2 a 2 verte (0, a) asmptote ( b, 0) (b, 0) verte asmptote (0, a) focus (0, -c) A hperbola with a vertical transverse ais has asmptotes with equations and. Eample: Sketch the hperbola with equation = 9 and find the vertices, foci, and asmptotes. 1. To write the equation in standard form, divide b 9. Summar of the Conic Sections 2. Because the 2 -term is positive, the transverse ais is horizontal. 3. Vertices: ( 3, 0), (3, 0) 4. Asmptotes: 5. (0, 1) a = 3 and b = 1 foci: (-3, 0) (0, -1) (3, 0) Summar of the Conic Sections Summar of the Conic Sections Conic sections presented in this chapter are of the form where either A or C must be nonzero and B = 0. Conic Section Characteristic Eample Circle Ellipse Either A = 0 or C = 0, but not both. AC > 0, A = C AC > 0, A C = 2 = Last 3 rows of the Table on pg 6-52 Hperbola AC < 0 9

10 Summar of the Conic Sections Conic sections presented in this chapter are of the form where either A or C must be nonzero and B 0. Conic Section Characteristic Eample Ellipse (or a Circle) Hperbola B 2-4AC = 0 B 2-4AC < 0 B 2-4AC > 0 Determining the tpe of the Conic Section To recognize the tpe of conic section, we ma need to transform the equation. Eample: Determine the tpe of conic section represented b each equation. (a) (b) (c) (d) Determining the tpe of the Conic Section (a) Hperbola centered at the origin. (b) Ellipse centered at (2, -3). (c) Circle centered at (4, -5), radius = 6. (d) with verte at (3, 2) and opens downward. STANDARD GRAPHS FORM OF OF EQUATIONS RATIONAL FUNCTIONS OF TRANSLATED CONICS In the following equations the point (h, k) is the verte of the parabola and the center of the other conics. CIRCLE ( h) 2 + ( k) 2 = r 2 Horizontal ais Vertical ais PARABOLA ( k) 2 = 4p( h) ( h) 2 = 4p( k) ELLIPSE a > b, c 2 = a 2 b 2 HYPERBOLA c 2 = a 2 + b 2 ( h) 2 ( k) 2 ( h) + = 1 2 ( k) 2 + = 1 a 2 b 2 b 2 a 2 ( h) 2 ( k) 2 ( k) 2 ( h) 2 = 1 a 2 b 2 = 1 a 2 b 2 CLASSIFYING A CONIC FROM ITS EQUATION CLASSIFYING A CONIC FROM ITS EQUATION The equation of an conic can be written in the form CONCEPT SUMMARY CONIC TYPES A 2 + B + C 2 + D + E + F = 0 which is called a general second-degree equation in and. The epression B 2 4AC is called the discriminant of the equation and can be used to determine which tpe of conic the equation represents. The tpe of conic can be determined as follows: DISCRIMINANT (B 2 4AC) TYPE OF CONIC < 0, B = 0, and A = C Circle < 0, and either B 0, or A C Ellipse = 0 > 0 Hperbola If B = 0, each ais is horizontal or vertical. If B 0, the aes are neither horizontal nor vertical. 10

11 Classifing a Conic Classif the conic = 0. SOLUTION Since A = 2, B = 0, and C = 1, the value of the discriminant is: B 2 4AC = 0 2 4(2)(1) = 8 Because B 2 4AC < 0 and A C, the graph is an ellipse. Classifing a Conic Classif the conic = 0. SOLUTION Since A = 4, B = 0, and C = 9, the value of the discriminant is: B 2 4AC = 0 2 4(4)( 9) = 144 Because B 2 4AC > 0, the graph is a hperbola. CLASSIFYING A CONIC FROM ITS EQUATION CLASSIFYING A CONIC FROM ITS EQUATION CONCEPT SUMMARY CONIC TYPES CONCEPT SUMMARY CONIC TYPES Back The tpe of conic can be determined as follows: Back The tpe of conic can be determined as follows: DISCRIMINANT (B 2 4AC) TYPE OF CONIC DISCRIMINANT (B 2 4AC) TYPE OF CONIC < 0, B = 0, and A = C Circle < 0, B = 0, and A = C Circle < 0, and either B 0, or A C Ellipse < 0, and either B 0, or A C Ellipse = 0 = 0 > 0 Hperbola > 0 Hperbola If B = 0, each ais is horizontal or vertical. If B 0, the aes are neither horizontal nor vertical. If B = 0, each ais is horizontal or vertical. If B 0, the aes are neither horizontal nor vertical. 11