Pre-Calc Name: ______UNIT 4: Conic Sections – REVIEW #2 Date: ______

THE CIRCLE STANDARD FORM GENERAL FORM (풙 − 풉)ퟐ + (풚 − 풌)ퟐ = 풓ퟐ 풙ퟐ + 풚ퟐ + 푫풙 + 푬풚 + 푭 = ퟎ Center: (ℎ, 푘) Radius: 푟 Complete the square to get Standard Form

THE PARABOLA GENERAL FORM 풙ퟐ + 푫풙 + 푬풚 + 푭 = ퟎ 풚ퟐ + 푫풙 + 푬풚 + 푭 = ퟎ Complete the square to get Standard Form STANDARD FORM (풙 − 풉)ퟐ = ퟒ풑(풚 − 풌) (풚 − 풌)ퟐ = ퟒ풑(풙 − 풉) Vertex: (ℎ, 푘) Opening: Vertex: (ℎ, 푘) Opening: Focus: (ℎ, 푘 + 푝) UP if 푝 is positive Focus: (ℎ + 푝, 푘) RIGHT if 푝 is positive Directrix: 푦 = 푘 − 푝 DOWN if 푝 is negative Directrix: 푥 = ℎ − 푝 LEFT if 푝 is negative

THE ELLIPSE GENERAL FORM 푨풙ퟐ + 푪풚ퟐ + 푫풙 + 푬풚 + 푭 = ퟎ where 퐴 ≠ 퐶 Complete the square to get Standard Form (NOTE: Make sure to factor first!) STANDARD FORM (풙 − 풉)ퟐ (풚 − 풌)ퟐ (풙 − 풉)ퟐ (풚 − 풌)ퟐ + = ퟏ + = ퟏ 풂ퟐ 풃ퟐ 풃ퟐ 풂ퟐ Major Axis: 2푎 Foci: ±푐 Major Axis: 2푎 Foci: ±푐 푐2 = 푎2 − 푏2 푐2 = 푎2 − 푏2 Minor Axis: 2푏 Minor Axis: 2푏 푎2 > 푏2 푎2 > 푏2

THE HYPERBOLA GENERAL FORM 푨풙ퟐ − 푪풚ퟐ + 푫풙 + 푬풚 + 푭 = ퟎ where 퐴 ≠ 퐶 푨풚ퟐ − 푪풙ퟐ + 푫풙 + 푬풚 + 푭 = ퟎ where 퐴 ≠ 퐶 Complete the square to get Standard Form (NOTE: Make sure to factor first!) STANDARD FORM (풙 − 풉)ퟐ (풚 − 풌)ퟐ (풚 − 풌)ퟐ (풙 − 풉)ퟐ − = ퟏ − = ퟏ 풂ퟐ 풃ퟐ 풂ퟐ 풃ퟐ 푏 푎 Asymptotes: 푦 − 푘 = ± (푥 − ℎ) Asymptotes: 푦 − 푘 = ± (푥 − ℎ) 푎 푏 Transverse Axis: 2푎 Foci: 푐 Transverse Axis: 2푎 Foci: 푐 Conjugate Axis: 2푏 푐2 = 푎2 + 푏2 Conjugate Axis: 2푏 푐2 = 푎2 + 푏2 푎2 is not necessarily larger than 푏2 푎2 is not necessarily larger than 푏2 Directions: State the conic section of the graph

1.) 16푥2 − 푦2 + 96푥 + 8푦 = −112 Circle Parabola Ellipse Hyperbola

2.) 7푥2 + 3푦2 − 28푥 − 12푦 = −19 Circle Parabola Ellipse Hyperbola

3.) 푥2 + 푦 + 4푥 − 2 = 0 Circle Parabola Ellipse Hyperbola

4.) 푥2 + 푦2 − 푥 + 2푦 − 12 = 0 Circle Parabola Ellipse Hyperbola

5.) 푦2 − 4푥2 − 2푦 − 16푥 + 1 = 0 Circle Parabola Ellipse Hyperbola

6.) 푥2 − 3푦2 + 6푦 + 6푥 = 18 Circle Parabola Ellipse Hyperbola

7.) 푥2 + 3푦2 + 2푥 − 5푦 − 129 = 0 Circle Parabola Ellipse Hyperbola

8.) 2푥2 − 5푥 + 푦 − 19 = 0 Circle Parabola Ellipse Hyperbola

9.) 5푥2 + 2푦2 − 2푥 − 9푦 − 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.) (푥 − 2)2 + 푦2 = 49 20.) (푥 + 3)2 + (푦 − 5)2 = 48 21.) (푥 − 7)2 + (푦 + 2)2 = 45

Center: ______Center: ______Center: ______

Radius: ______Radius: ______Radius: ______

Directions: Find the vertex, focus, directrix and 풑 value for each parabola. Then graph each parabola.

22.) (푥 + 2)2 = 16(푦 − 1) 23.) (푦 + 1)2 = 8(푥 + 4) 24.) (푥 + 2)2 = −4(푦 − 3)

Vertex: ______Focus: ______Vertex: ______Focus: ______Vertex: ______Focus: ______

Directrix: ______푝 = ______Directrix: ______푝 = ______Directrix: ______푝 = ______

Directions: Change the following from General Form to Standard Form and find the center and radius. 25.) 푥2 + 푦2 − 4푥 + 2푦 + 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. 26.) (푥 − 2)2 = 12푦 + 36

Vertex: ______Focus: ______

푝 = ______Directrix: ______

Directions: Convert the general form of the equation of the ellipse into standard form. 27.) 9푦2 + 108푦 + 4푥2 − 56푥 + 484 = 0

Standard form: ______

Directions: Convert the general form of the equation of the hyperbola into standard form. 28.) 25푥2 − 9푦2 − 100푥 − 72푦 − 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.) 25푥2 + 100푥 + 4푦2 + 8푦 + 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.) 푦2 − 4푦 − 4푥2 + 40푥 − 100 = 0

Center: ______

Foci: ______

______

Vertices: ______

______

Asymptotes: ______

Length of Transverse Axis: ______

Length of Conjugate Axis: ______