Notes 11 Key - Teacher

NOTES 11: ANALYTIC GEOMETRYName:______________________________Date:________________Period:_________Mrs. Nguyen’s Initial:_________________LESSON 11.1 – PARABOLASGeometricDefinition of aParabolaA parabola is the set of points in the planeequidistant from a fixed point F (calledthe focus) and a fixed line l (called thedirectrix).Drawing:Parabola withVertical Axis(Opens up or down)2( x h) 4 p( yk)Focus: hk, pDirectrix: y kpDrawing:Parabola withHorizontal Axis(Opens left or right)2( y k) 4 p( xh)Focus: hp,kDirectrix: x hpDrawing:Note: The distance between the vertex and focus is p units and the distance between the vertex and directrix is punits.Practice Problems: Find the focus, directrix, & focal diameter of the parabola. Then graph the parabola.1.2y 4x 8x 62.2 2x y 6y123Mrs. Nguyen – Honors Algebra II – Chapter 11 Notes – Page 1

LESSON 2.2 – GRAPHS OF EQUATIONS IN TWO VARIABLESStandard form of aCircleThe point (x, y) lies on the circle of radius r and2 2 2x h yk rcenter (h, k) iff Center:Radius:Practice Problems: Write the standard form for the following circles.1. The point (3, 4) lies on a circle whose center isat (-1, 2).2. Endpoints of a diameter: (-4, -1), (4, 1).Practice Problems: Complete the square, if necessary, to find the center and radius. Then draw the graph.3.2 2x y 8x6y 04.2 2x y 4x805.2 2x y 2x4y11 06.2 2x y 6x2y150Mrs. Nguyen – Honors Algebra II – Chapter 11 Notes – Page 3

LESSON 11.2 – ELLIPSESGeometricDefinition of anEllipseAn ellipse is the set of all points in theplane, the sum of whose distances fromtwo fixed points F and F is a constant.12These two fixed points are the foci (pluralof focus) of the ellipse.Drawing:Equation of anEllipse Centered at(h, k):2 2( xh) ( yk)2 2a b 12 2( xh) ( yk)2 2b aHorizontal Major AxisVertical Major AxisThe variable with the largest denominator tells you which direction the ellipse goes. “a” is always the biggernumber in ellipses.a = distance from the center to each vertexb = distance from the center to each co-vertex. 1Major Axis length = 2aThe Foci of anEllipseThe Eccentricity ofan EllipseMinor Axis length = 2bThe foci of the ellipse lie on the major axis, c units form the center whereThe eccentricity, e, determines how round the ellipse is:ce .a2 2 2c a b .Practice Problems: Find the center, vertices, foci, & eccentricity of the ellipse. Determine the lengths of themajor and minor axes, and sketch the graph.1.2 24x8y 322.2 2x 25y 8x100y91 0Mrs. Nguyen – Honors Algebra II – Chapter 11 Notes – Page 4

LESSON 11.3 – HYPERBOLASGeometricDefinition of aHyperbolaA hyperbola is the set of all points in theplane, the difference of whose distancesfrom two fixed points F and F is a12constant. These two fixed points are thefoci (plural of focus) of the hyperbola.Drawing:Equation of aHyperbolaCentered at (h, k):2 2( xh) ( yk)2 2a b 1bAsymptotes: y k ( xh)aHorizontal Transverse Axis: Thehyperbola opens in the x-direction if thesign in front of the term containing x ispositive.2 2( yk) ( xh)2 2a b 1aAsymptotes: y k ( xh)bVertical Transverse Axis: The hyperbolaopens in the y-direction if the sign in frontof the term containing y is positive.“a” is always the number under the positive variable.Transverse axis length = 2aThe Foci of aHyperbolaThe foci of the hyperbola lie c units from the center where2 2 2c a b .Practice Problems: Find an equation for the hyperbola whose center is at the origin that satisfies the givenconditions.1. Foci: 0, 10, vertices: 0, 82. Vertices: 0, 6, asymptotes:1y x33. Foci 6,0, vertices 2,04. Foci 3, 0 , hyperbola passes through (4, 1)Mrs. Nguyen – Honors Algebra II – Chapter 11 Notes – Page 6

7.2x 8x4y4 08.2 236x y 216x2y289 0Practice Problems: Write the equation of the conic described in standard form.9. Write the equation of the ellipse with a centerof (2, −4) and a co-vertex at (2, −1) and a vertex at (7,−4).10. Write the equation of the vertical parabola witha vertex of (6, 1) and contains the point (4, 5).11. Write the equation of an ellipse with foci of (4,0) and (−4, 0) and co-vertices of (0, 2) and (0, −2).12. A parabola with a focus of (5, 6) and directrixof x = 1.13. A hyperbola with vertices at (4, -2) and (4, -6)and foci at (4, 1) and (4, -9).14. A hyperbola with vertices at (2, 0) and (6, 0)and one asymptote whose equation is y = 2x – 8Mrs. Nguyen – Honors Algebra II – Chapter 11 Notes – Page 9