Lecture Note
University
California State UniversityCourse
MATH 150B | Calculus IIPages
5
Academic year
2023
yung dump
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11.6 Conic Sections 91 Conic sections are Parabolas Ellipses, Hyperbolas curves (1) (2) (3) in which a plane cuts a double cone. (3) (1) (2 Parabolas: The standard form y 1 x2=ypy Focus of the Parabola is p(-xi) F(e,P) p(x,y) x = y py , P>0 Vertex Q(-x,P) Q(XIP) pirectrix y=-p the Parabola is the set of points s.t PF = PQ (x-o)*+(y-p)2 = (x-x)2 +(y--p)2 x2+y-p)2 = (y+p)2 x2+(y-p)2 = (2+p)2 x yyy P is positive number called the parabola's foral length. 2 when x2 -ypy, P>O the parabola opens down: when y2-4px, P>O Directrix y=p 3 Vertex the parabola opens right x 4 when y' = - -ypx, p>o Foas Flo,-P the parabola opens left x-ypy Directrix y 3-ypx ya= y pirectrix vertex vertex F(P,0) X F(-P,O) x X=-P X=P
Exp Find the focus and directrix for each of the 92 following parabolas. sketch each one. y y=12x 1 y'= 12X vertex is (0,0) 4P = 12 (> p=3 / is (3,0) Directrix is x=-3 F(3,0) x Focus X=-3 2 y'= - 2x p=1/2 / vertex is (0,0) up= 2 Focus is (-1/20), Directrix is x= 1/2 F 3 x'= - 8y stext x YP=8 p=2 *= 1/2 Focus is (0,-2) y Directrix is y=2 y= 2 Vertex is (0,0) x 4 y=4x2 F(0,-2) 2 x2 X -sy = y 4P = 1 p=16 y y=yx2 Focus is (0,16) (0,16) Directrix is y= 16 y= 16 x Vertex is (0,0) 1y b Ellipse P(x,y) vertex F1 center F2 vertex F F2 Focus Focus a X (-c,o) (c,o) Focal axis The ellipse defined by the equation -b PF, t P F2 = 2a is the graph of a>b x2 + y = 1 where c-= 2 b 2 is the Center to- focus distance. a2 b2 a is the semimajor axis b is the semiminor axis
93 * Major axis of the ellipse is the line segment of length 2a Minor axis of the ellipse is the line segment of length zb. standard-Form Equations for Ellipse Centered at the Origin : y Foci on the X- axis: x2 2 =1 b + a>b a2 b2 (-C,O) (c,o) Center-to-focus distance c=va?-ba F1 F2 x a Foci: (+c,o) -b vertices : ( + a, 01 y * Foci on the y- axis : x2 + y2 = b2 a2 a>b b x Center-to-focus distance C= a?-b2 (o,-c).F, Foci: ( 0, cc) -a Vertices : ( 0, + a) ifa=b C=0 we get a circle Exp put each of the following equations in the standard form. sketch the ellipse and include the foci. V2 I 2 x2 + y = 2 F2 vertices : (0,IV2) x2 y2 + 2 Foci: (0,41) -1 X C= a?-b2 2 = 2-1=1 -10F 2 9x2 10 y2 y + = 90 2 X y superscript(2) 2 Vertices (5510,0) + 10 9 Focis (11,0) F, F2 a? b2 = V10-9=1 -VIO -1 I VIO X C =
94 Hyperbolas vertices Focal 2(x,9) F, b F2 axis (-c10) (c10) Focus center focus o F1 a a F2 x The hyper bola defined by the equation PF, - PF2 = 2a is the graph of as x2 y = I where c=Va b2 standard form Fquations for Hyperbolas Centered at Origin: y Foci on the x-axis : x2 y=1 a2 b2 (-C,O) (c,o) Center-to- focus distance F1 F2 x C= V a2+bz Foci: Vertices: (ta, o) y Asymptotes: y = I bl X F2(CO,C) * Foci on the y- - axis : y 2 - x2 = I a2 b2 Center- to- focus distance C x = a?+b2 Foci: (o, + c) F1 (e,-c) Vertices : (o, ta) Asymptoles : y = + a b X Exp Put each of the following equations in the standard form. II 8x2-2y2 16 (=) x2 2 sketch the hyperhold y2 and include the faci and asymptotes. a=V2 2 y2-3x2=3 G> y2 x = 1 C= az+b? = 2+8 =VTO b=2h 3 C= Asymptotes y== F3x Asymntoles y==2x 1== y F2 (0,2) 53 F1 F2 -Vn X r (V10,0) -v3 x (-V10,0) Fit (0,-2)
Exp show that x2- represents a hyperbola. 95 Find it's center, asymptotes and foci. 3 F1 = (x-1)2-(y-2)2=1 2 X center (1,2) c= a'+b2 =V2 foci : Asymplotes y-2=+ (x-1) vertices: (0,2) and (2,2) (1+9,2) (y-3)2 (x-1) 3 Center (1,3) Vertices foci : (1,3tc) , c=Vaz+b2 (1,9,(1,5) = V3+1=2 Asymptotes Y-3==V3(x-1)
Conic Sections
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