Parabola Ellipse and Hyperbola

Pre-Calc UNIT 4: Conic Sections – REVIEW #2 Name: __________________________ Date: _________________________ THE CIRCLE STANDARD FORM GENERAL FORM (𝒙 − 𝒉)𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓𝟐 𝒙𝟐 + 𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 Center: (ℎ, 𝑘) Radius: 𝑟 Complete the square to get Standard Form THE PARABOLA GENERAL FORM 𝟐 𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 𝒙 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 Complete the square to get Standard Form STANDARD FORM (𝒙 − 𝒉)𝟐 = 𝟒𝒑(𝒚 − 𝒌) Vertex: (ℎ, 𝑘) (𝒚 − 𝒌)𝟐 = 𝟒𝒑(𝒙 − 𝒉) Vertex: (ℎ, 𝑘) Opening: Focus: (ℎ, 𝑘 + 𝑝) UP if 𝑝 is positive DOWN if 𝑝 is negative Directrix: 𝑦 = 𝑘 − 𝑝 Opening: Focus: (ℎ + 𝑝, 𝑘) RIGHT if 𝑝 is positive LEFT if 𝑝 is negative Directrix: 𝑥 = ℎ − 𝑝 THE ELLIPSE GENERAL FORM 𝑨𝒙𝟐 + 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 where 𝐴≠𝐶 Complete the square to get Standard Form (NOTE: Make sure to factor first!) STANDARD FORM (𝒙 − 𝒉)𝟐 (𝒚 − 𝒌)𝟐 + =𝟏 𝒂𝟐 𝒃𝟐 Major Axis: 2𝑎 Foci: ±𝑐 (𝒙 − 𝒉)𝟐 (𝒚 − 𝒌)𝟐 + =𝟏 𝒃𝟐 𝒂𝟐 Major Axis: 2𝑎 Foci: ±𝑐 𝑐 2 = 𝑎2 − 𝑏 2 𝑎2 > 𝑏 2 Minor Axis: 2𝑏 𝑐 2 = 𝑎2 − 𝑏 2 𝑎2 > 𝑏 2 Minor Axis: 2𝑏 THE HYPERBOLA GENERAL FORM 𝑨𝒙𝟐 − 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 where 𝐴≠𝐶 𝑨𝒚𝟐 − 𝑪𝒙𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 where 𝐴≠𝐶 Complete the square to get Standard Form (NOTE: Make sure to factor first!) STANDARD FORM (𝒙 − 𝒉)𝟐 (𝒚 − 𝒌)𝟐 − =𝟏 𝒂𝟐 𝒃𝟐 Transverse Axis: 2𝑎 𝑏 𝑦 − 𝑘 = ± (𝑥 − ℎ) 𝑎 Foci: 𝑐 Conjugate Axis: 2𝑏 𝑐 2 = 𝑎2 + 𝑏 2 Asymptotes: 𝑎2 is not necessarily larger than 𝑏 2 (𝒚 − 𝒌)𝟐 (𝒙 − 𝒉)𝟐 − =𝟏 𝒂𝟐 𝒃𝟐 Asymptotes: Transverse Axis: 2𝑎 Conjugate Axis: 2𝑏 𝑎 𝑦 − 𝑘 = ± (𝑥 − ℎ) 𝑏 Foci: 𝑐 𝑐 2 = 𝑎2 + 𝑏 2 𝑎2 is not necessarily larger than 𝑏 2 Directions: State the conic section of the graph 1.) 16𝑥 2 − 𝑦 2 + 96𝑥 + 8𝑦 = −112 Circle Parabola Ellipse Hyperbola 2.) 7𝑥 2 + 3𝑦 2 − 28𝑥 − 12𝑦 = −19 Circle Parabola Ellipse Hyperbola 3.) 𝑥 2 + 𝑦 + 4𝑥 − 2 = 0 Circle Parabola Ellipse Hyperbola 4.) 𝑥 2 + 𝑦 2 − 𝑥 + 2𝑦 − 12 = 0 Circle Parabola Ellipse Hyperbola 5.) 𝑦 2 − 4𝑥 2 − 2𝑦 − 16𝑥 + 1 = 0 Circle Parabola Ellipse Hyperbola 6.) 𝑥 2 − 3𝑦 2 + 6𝑦 + 6𝑥 = 18 Circle Parabola Ellipse Hyperbola 7.) 𝑥 2 + 3𝑦 2 + 2𝑥 − 5𝑦 − 129 = 0 Circle Parabola Ellipse Hyperbola 8.) 2𝑥 2 − 5𝑥 + 𝑦 − 19 = 0 Circle Parabola Ellipse Hyperbola 9.) 5𝑥 2 + 2𝑦 2 − 2𝑥 − 9𝑦 − 22 = 0 Circle Parabola Ellipse Hyperbola Directions: Write the equation of each graph in standard form. 10.) _________________________ 11.) _________________________ 12.) _________________________ 13.) _________________________ 14.) _________________________ 15.) _________________________ 16.) _________________________ 17.) _________________________ 18.) _________________________ Directions: Find the center and the radius of each circle: 19.) (𝑥 − 2)2 + 𝑦 2 = 49 20.) (𝑥 + 3)2 + (𝑦 − 5)2 = 48 21.) (𝑥 − 7)2 + (𝑦 + 2)2 = 45 Center: __________ Center: __________ Center: __________ Radius: __________ Radius: __________ Radius: __________ Directions: Find the vertex, focus, directrix and 𝒑 value for each parabola. Then graph each parabola. 22.) (𝑥 + 2)2 = 16(𝑦 − 1) Vertex: _______ Directrix: _______ 23.) (𝑦 + 1)2 = 8(𝑥 + 4) Focus: _______ Vertex: 𝑝= _______ _______ Directrix: _______ 24.) (𝑥 + 2)2 = −4(𝑦 − 3) Focus: _______ Vertex: 𝑝= _______ _______ Directrix: _______ Focus: _______ 𝑝= _______ Directions: Change the following from General Form to Standard Form and find the center and radius. 25.) 𝑥 2 + 𝑦 2 − 4𝑥 + 2𝑦 + 1 = 0 Standard Form: ________________________________________ Center: ____________ Radius: ____________ Directions: Find the coordinates of the vertex and the focus. Then find the equation of the directrix and sketch a graph for the following parabolas. 2 26.) (𝑥 − 2) = 12𝑦 + 36 Vertex: ____________ Focus: ____________ 𝑝= ____________ Directrix: ____________ Directions: Convert the general form of the equation of the ellipse into standard form. 27.) 9𝑦 2 + 108𝑦 + 4𝑥 2 − 56𝑥 + 484 = 0 Standard form: ______________________________ Directions: Convert the general form of the equation of the hyperbola into standard form. 28.) 25𝑥 2 − 9𝑦 2 − 100𝑥 − 72𝑦 − 269 = 0 Standard form: ______________________________ Directions: Find the coordinates of the center, foci, and vertices of the ellipse and the length of the major and minor axis. Then graph the equation. 29.) 25𝑥 2 + 100𝑥 + 4𝑦 2 + 8𝑦 + 4 = 0 Center: _______________________ Foci: _______________________ _______________________ Length of major axis: ________________ Length of minor axis: ________________ Major axis vertices: ________________ ________________ Minor axis vertices: _____________ _____________ Directions: Find the coordinates of the center, the foci, and the vertices, the length of the transverse and conjugate axis, and the equations of the asymptotes of the hyperbola. Then graph the equation. 30.) 𝑦 2 − 4𝑦 − 4𝑥 2 + 40𝑥 − 100 = 0 Center: ____________________ Foci: ____________________ ____________________ Vertices: ____________________ ____________________ Asymptotes: ____________________ Length of Transverse Axis: _______ Length of Conjugate Axis: _______