Unit 4 Conics Review Key

Pre-Calc UNIT 4: Conic Sections-REVIEW #2 Name: Key Date THE CIRCLE STANDARD FORM Center (h, k) Radius: r x ^ 2 + Dx + Ey + F = 0 GENERAL FORM x ^ 2 + y ^ 2 + Dx + Ey + F = 0 Complete the square to get Standard Form THE PARABOLA GENERAL FORM Complete the square to get Standard Form STANDARD FORM (x - h) ^ 2 = 4p(y - k) (y - k) ^ 2 = 4p(x - h) Vertex: (h, k) Opening: Vertex: (h,k) Opening: Focus: (h,k+p) UP if p is positive DOWN if p is negative Focus: (h+p,k) Directrix: yk-p Directrix: xh-p RIGHT if p is positive LEFT if p is negative THE ELLIPSE GENERAL FORM Ax² + Cy² + Dx+Ey+F=0 where AC Complete the square to get Standard Form (NOTE: Make sure to factor first!) STANDARD FORM Major Axis: 24 Fock tc Major Axis: 24 Foci c c ^ 2 = a ^ 2 - b ^ 2 a ^ 2 >= b ^ 2 Minor Axis: 26 a ^ 2 > b ^ 2 c ^ 2 = a ^ 2 - b ^ 2 Minor Axis: 25 THE HYPERBOLA GENERAL FORM Ax²-Cy² + Dx+Ey+F=0 where AC AyCx+Dx+Ey+F=0 where A C Complete the square to get Standard Form (NOTE: Make sure to factor first!) STANDARD FORM b y-k=-(x-h) a Asymptotes: Asymptotes: y-k=(x-h) Transverse Axis: 20 Fock c Transverse Axis: 20 Foci c Conjugate Axis: 26 a is not necessarily larger than b Conjugate Axis: 26 a is not necessarily larger than b y ^ 2 + Dx + Ey + F = 0 (x - h) ^ 2 + (y - k) ^ 2 = r ^ 2 c ^ 2 = a ^ 2 + b ^ 2 c ^ 2 = a ^ 2 + b ^ 2 ((x - h) ^ 2)/(a ^ 2) + ((y - k) ^ 2)/(h ^ 2) = 1 ((y - k) ^ 2)/(a ^ 2) - ((x - h) ^ 2)/(b ^ 2) = 1 ((x - h) ^ 2)/(b ^ 2) + ((y - k) ^ 2)/(a ^ 2) = 1 ((x - k) ^ 2)/(a ^ 2) - ((y - k) ^ 2)/(b ^ 2) = 1 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. e ( 1) 2 a = 3 (2) 10.) (x + 1) ^ 2 + (y - 2) ^ 2 = 9 ) (x + 2) ^ 2 = 8(y + 3) 12.) (y - 1) ^ 2 = - 4(x + 4) ( 1 j^ -2) ( 1)+(12) 13.) 9 a = 5 b = 3 2- (21) (- 2, - 2) 4 14. ((x - 2) ^ 2)/9 + ((y - 1) ^ 2)/25 = 1 15.) ((x + 1) ^ 2 + (y + 2) ^ 2)/4 = 1 b = 2 a = 4 (14,1) p = 5 a = 4 b - 3 (- 1, 1) 03 62 (2,1) (- 1, 1) alpha = 3 b = 4 16.) (x + 1) ^ (2 ^ 0) -((y + 1) ^ 2)/9 = 1 17.) (y ^ - 1) ^ 2 - ((x ^ - 2) ^ 2)/4 = 1 18.) )x+1(-)-1(ا 9 116 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: (2, 0) Center: (-3,5) Center: ( 7 ,*2) Radius: 7 Radius: sqrt(45) Radius: sqrt(45) Directions: Find the vertex, focus, directrix and p value for each parabola. Then graph each parabola. 22.) (x + 2) ^ 2 = 16(y - 1) 23.) (y + 1) ^ 2 = 8(x + 4) 24.) (x + 2) ^ 2 = - 4(y - 3) Vertex: (2,1) Focus: (-2,5) Vertex: (-4,1) Focus: (-2, 1) Vertex: (-2,3) Focus: (2,2) Directrix: y = - 3 p = 4 Directrix: x = - 6 p = 2 Directrix: y = 4 6 p = -1 + +