University:
California State University, Northridge
Course:
MATH 150B | Calculus II
Academic year:

2023
Views:

374

Pages:

5
Author:

Yung D

11.6 Conic Sections 91 * Conic Sections are Parabolas, Ellipses, Hyper belas in which a plane cuts a double cone. Parabolas: III The standard form of the parabola is x=ypy Pl-xi F(0,P). P(x,y) Focus Vefter (-x,p) Pirectrix y=-p the Parabola is the set of points sit PF = PQ √(x-0)+(y-p)² = √(x-x)+(--) √x²+(y-p) = √(+) x²+(y-p)²= (+p)²²=ypy 2 P is positive number called the parabola's focal length. 2) when x²= -4py, pro the parabola opons down: √ 3 when y²=4px, p>0 the parabola opens right Pirectrix y y=p Vertex Foas F(-) Pirectrix [ when y=-4px, p>0 the parabola opens left Directrix ny y=ypx verter F(P10) ← X F(-1,0) Vortex x= P Exp Find the focus and directrix for each of following parabolas. sketch each one. y= 12X 4P=12 P=3 vertex is (0,0) Focus is (3,0), Directrix is x=-3 2 y=-2x 4P=2=, vertex is (0,0) Focus is (-10), Directrix is x=± x=-8y 4P=8P=2 Focus is (0,-2) Pirectrix is y=2 Vertex is (0,0) y = 4 x² x² = + y 4P = P= Focus is (o, to) Directrix is y = -to Vertex is (0,0) ** Major axis of the ellipse is the line seg ment of length za 93 ** Minor axis of the ellipse is the line segment of length zb. Standard-Form Equations for Ellipse Centered at the Origin: * Foci on the x-axis: X al L + y be as b Center-to-focus distance c=va²-b² • Foci: (+c, 0) • Vertices: (±a, ol * Foci on the y-axis X + •Center-to-focus distance = a²-b² c=Vab . Foci: (0,c) if abco we get •Vertices: (0,±a) a circle Exg Put each of the following equations in the standard form. Sketch the ellipse and include the foci. 2x²+y²=2 195 Focis (0,±1) c = √a²-b² = √2-1=1 ② 9x² + 10 y² = 90 2 2 c=√a²-b² =√10-9=1 Vertices: (0,1VE) Vertices (Irio, 0) Focis (21, 0) The hyper bola defined by the equation PF, PF₂ = 2 a is the graph of x2 ar where c=Va²+b² standard form Equations for Hyperbolas Centered at Origin: * Foci on the x-axis: -=1 • Center-to-focus distance C= Va² + b² Foci: (±c,0) • Vertices: (ta, o) • Asymploles: y=x * Foci on the y-axis: • Center-to-focus distance c = Vaz+b² • Foci: (0, ±c) Vertices: (0, 1a) Asymptoles: y = ± Exp Put each of the following equations in the standard form. Sketch the hyperhda and include the foci and asymptotes 8x²-2y²=16=1 2y²-3x²=3 C= √3+1 = 2 C= √a²+b² = √2+8=√10 6=2√2 Asymploles y= ±2x Exp show that x²- y -2x+4y=4 Find it's center, asymptotes and foci. r 2 (x-1)-1-(y-2)+4= 4 (x-1)²- (y-2)²=1 center (1,2) <= √a²+b² = √2 foci: (1±2,2 • Asymptotes y-2=± (x-1) • Vertices: (0,2) and (2.2) :(11912) (2-3)-(x-1)² = 1 3 Center (1,3) • Vertices (1,3±a) = (1,3±√3) foci: (1,3±c) (1,1), (1,5) ,c=Va²+b² =√3+1=2 • Asymptoles y-3==√(x-1)