Math 4 Midterm 1 Review CONICS [1] Find the equations of the following conics. If the equation corresponds to a circle find its center & radius. If the equation corresponds to a parabola find its focus vertex directrix & axis of symmetry. If the equation corresponds to an ellipse find its center foci & the endpoints of the major and minor axes. If the equation corresponds to a hyperbola find its center foci vertices & asymptotes. [a] Ellipse: foci 5 and minor axis of length 1 Parabola: vertex 4 6 directrix x 16 Hyperbola: one focus 1 asymptotes y x and y x 5 [d] Ellipse: endpoints of major axis and 5 [e] Hyperbola: vertices 4 1 and 1 passing through 5 6 [f] Parabola: vertex 4 6 focus 4 11 [g] Ellipse: vertices and 5 foci 6 and 4 [h] Circle: endpoints of diameter 8 and [i] Hyperbola: vertices and 5 foci 9 and [j] Ellipse: foci 5 and 5 4 major axis of length 1 [k] Parabola: focus 4 6 directrix y 16 [l] Ellipse: vertices 1 1 and 1 minor axis of length 4 [m] Hyperbola: one vertex 6 asymptotes y x 4 and y x 8 [n] Parabola: vertex at origin horizontal axis of symmetry passing through 8 [o] Ellipse: endpoints of minor axis 11 and major axis of length 10 endpoints of minor axis 1 and 1 1 [] Determine if each equation corresponds to a circle a parabola an ellipse or a hyperbola. If the equation corresponds to a circle find its center & radius. If the equation corresponds to a parabola find its focus vertex directrix & axis of symmetry. If the equation corresponds to an ellipse find its center foci & the endpoints of the major and minor axes. If the equation corresponds to a hyperbola find its center foci vertices & asymptotes. [a] 9x 16y 6x y 164 0 y 8x y 9 0 x y 6x 8y 1 0 [d] 9x 4y x 40y 08 0 [] Draw diagrams and write algebraic equations involving distances to answer the following questions. POLAR [a] A car is traveling along a route so that it is always 40 miles closer to a certain cell tower than it is to a certain monument. What is the shape of the route? A ship is traveling along a route so that any signal originating at Old Port headed to the ship and then immediately resent to New Port always takes 8 seconds to complete the transmission. What is the shape of the route? A satellite is traveling in an orbit so that it is always 5 000 miles from the center of the earth. What is the shape of the orbit? [1] Remember that a single point in the plane has infinitely many polar co-ordinates. Consider the point with polar co-ordinates. [a] Find another pair of polar co-ordinates for this point using a positive r value and a positive value. Find another pair of polar co-ordinates for this point using a positive r value and a negative value. Find another pair of polar co-ordinates for this point using a negative r value and a positive value. [d] Find another pair of polar co-ordinates for this point using a negative r value and a negative value.
[] Convert the following points or equations. [a] 5 the point with polar co-ordinates 8 6 to rectangular co-ordinates the point with rectangular co-ordinates 6 to polar co-ordinates the rectangular equation x y x 0 to polar [d] 4 cos y 6 cos 6y 9 5 to rectangular 6 the polar equation r to rectangular [e] the rectangular equation x 0 to polar [f] the polar equation r to rectangular [g] the rectangular equation x 0 to polar [h] the polar equation [] Run the standard tests for symmetry for the polar equation r 1 sin and state the conclusions. What is the minimum interval of values that must be plotted before using symmetry to complete the graph? [4] Find the values of [ 0 at which the graph of the polar equation r cos 1 [5] Name the shape of the graphs of the following polar equations. If the graph is a rose curve state the number of petals. passes through the pole. [a] [e] [i] r 5 5sin r sin 6 r 5 [d] r 4sin 9 [f] r cos [g] r 6 4cos [h] r sin r 6 sin [6] Determine if each polar equation corresponds to a circle a parabola an ellipse or a hyperbola. If the equation corresponds to a circle find its center & radius. If the equation corresponds to a parabola find its eccentricity focus directrix & vertex. If the equation corresponds to an ellipse find its eccentricity foci directrix center & the endpoints of the major axes and latera recta. If the equation corresponds to a hyperbola find its eccentricity foci directrix center vertices & the endpoints of the latera recta. Do not convert the equations to rectangular co-ordinates. Final answers must be in rectangular co-ordinates. [a] 10 r sin 10 r cos 10 r [d] r 10 sin [] Find the polar equations of the following conics with their focus at the pole. [a] Parabola: directrix x Parabola: vertex Ellipse: eccentricity 4 directrix y 5 [d] Ellipse: vertices 4 0 and [e] Hyperbola: eccentricity 5 directrix x [f] Hyperbola: vertices and 15 [8] Draw diagrams and write algebraic equations involving distances to answer the following questions. A drinking fountain is 15 feet from the wall of a school building. [a] A cat is running on the school grounds so that it is always three times as far from the wall as it is from the fountain. What is the shape of the cat s path? A dog is running on the school grounds so that it is always three times as far from the fountain as it is from the wall. What is the shape of the dog s path? A chicken is running on the school grounds so that it is always as far from the wall as it is from the fountain. What is the shape of the chicken s path? [9] Sketch the graphs of the polar equations in [5][a] [f] [g] and [i] using the shortcut process shown in lecture. Find all y intercepts. x and
ANSWERS CONICS [1] [a] Center: 1 Endpoints of major axis: 1 1 Endpoints of minor axis: 1 4 and 1 8 x 1 5 y 6 1 Focus: 4 6 Vertex: Axis of Symmetry: y 6 y 6 80 x 4 Center: 1 1 and 10 Vertices: 1 5 Asymptotes: y 1 x 1 [d] Center: 1 1 5 Endpoints of major axis: Endpoints of minor axis: 18 5 x 1 6 1 5 y 16 [e] Center: 1 1 1 6 1 Vertices: Asymptotes: y 1 x 1 x 1 9 y 1 6 1 [f] Focus: Vertex: y 1 Axis of Symmetry: x 4 x 4 0 y 6 [g] Center: 1 Endpoints of major axis: Endpoints of minor axis: 11 1 1 x 11 y 1 6
[h] Center: 5 Radius: 4 x 5 y 4 [i] Center: 1 Vertices: Asymptotes: y x y 1 6 1 x 8 1 [j] Center: 5 1 Endpoints of major axis: 5 5 and 5 Endpoints of minor axis: 5 1 x 5 y 1 6 1 [k] Focus: Vertex: 4 5 Axis of Symmetry: x 4 x 4 44 y 5 [l] Center: 1 1 1 Endpoints of major axis: Endpoints of minor axis: 1 and x 5 1 [m] Center: 5 Vertices: 0 Asymptotes: y 1 4 and 6 x 9 1 49 [n] Focus: 0 Vertex: Axis of Symmetry: 0 49 x y 49 y 8 y 6 x
[o] Center: 9 1 9 Endpoints of major axis: 8 9 Endpoints of minor axis: and 9 x 5 y 9 4 1 [] [a] HYPERBOLA Center: 1 4 Vertices: and 6 and 4 y x 1 4 Asymptotes: [d] PARABOLA Focus: 1 Vertex: 5 1 x Axis of Symmetry: y 1 CIRCLE Center: 4 Radius: ELLIPSE Center: 4 5 4 5 5 Endpoints of major axis: 4 8 Endpoints of minor axis: 6 5 and 4 and 5 [] [a] part of hyperbola ellipse circle POLAR 8 [1] [a] 4 5 [d] [] [a] 4 4 [e] [g] 10.46 cos r cos sec [d] 1x 16y 8x 49 0 cos 6 r [f] x y x y sin cos r or [h] y x 1 sin 1 sin [] Symmetry over polar axis: substituting substituting Symmetry over pole: substituting r Symmetry over substituting r gives r 1 sin r gives r 1 sin gives r 1 sin r gives r 1 sin : substituting r gives r 1 sin substituting Minimum interval [ 0 ] r gives r 1 sin or ] [ symmetric over pole
[4] 0 cos 1 for 0 1 cos 4 8 10 4 5 since 0 4 [5] [a] cardioid rose curve with 1 petals circle [d] line [e] rose curve with 9 petals [f] limacon with inner loop [g] limacon with dimple [h] circle [i] convex limacon [6] [a] PARABOLA Eccentricity: 1 Focus: 0 0 y 10 Vertex: 0 5 ELLIPSE Eccentricity: 0 0 x 5 Center: 4 0 Endpoints of major axis: 0 Endpoints of latera recta: 0 10 and 8 0 and 10 0 and 8 10 HYPERBOLA Eccentricity: 0 0 Center: Vertices: Endpoints of latera recta: and 01 y 10 0 6 0 and 010 5 0 and 51 [d] Center: 0 0 Radius: 10 [] [a] [d] r 1 cos 8 r cos [e] 14 r 1 sin 15 r 5cos [f] 15 r 4 sin 15 r sin [8] [a] ellipse part of hyperbola parabola [9] [a] r 5 5sin [f] r cos [g] r 6 4cos [i] r 6 sin 5 0 5 0 1 0 0 0 0 10 0 6 0 0 0 0 10 0 0 6 0 8 0 4