Test Dates Thursday, January 4 Chapter 6 Team Test Friday, January 19 Chapter 9 Quiz Thursday, January 25 or Friday, January 26 Final Exam

Review Circles: 1. Find the center and radius of the circle. 𝒙 𝟐 +𝟏𝟎𝒙+ 𝒚 𝟐 −𝟏𝟒𝒚+𝟒𝟎=𝟎 2. Find an equation of the circle having (2,5) and (-2,-1) as endpoints of the diameter. Ellipses: 3. Find the coordinates of the vertices and the foci, and sketch the ellipse with equation 9 𝒙 𝟐 + 𝟓𝒚 𝟐 =𝟒𝟓 4. Find an equation of the ellipse that has (0,-4) and (0,4) as vertices and (-3,0) and (3,0) as endpoints of its minor axis. Hyperbolas: 5. Find an equation for the hyperbola that has center at (0,0), a vertex at (0,-3), and a focus at 𝟎,− 𝟏𝟑 . 6. Find an equation for the hyperbola that has a vertex at (2,0) and asymptotes with equations 𝒚=±𝟐𝒙. Parabolas: 7. Find the coordinates of the vertex and the focus, and the equation of the directrix, of the parabola 𝐲= 𝟏 𝟔 𝒙 𝟐 8. Write an equation of the parabola whose directrix is 𝒙=−𝟑 and whose focus is (3,0) Systems: 9. Solve the system algebraically to find the intersection points. Then graph the system. 𝒙 𝟐 + 𝒚 𝟐 =𝟐𝟓 𝒙 𝟐 + 𝟏𝟎𝒚 𝟐 =𝟏𝟔𝟗 Work for 10 mins on 2 problems, then 5 min review. 1 hr, 15 mins

1. Find the center and radius of the circle. 𝑥 2 +10𝑥+ 𝑦 2 −14𝑦+40=0 𝑥 2 +10𝑥+25+ 𝑦 2 −14𝑦+49=−40+25+49 𝑥+5 2 + 𝑦−7 2 =34 Center = (−5,7) Radius = 34

𝐷𝑖𝑠𝑡= 𝑥 2 − 𝑥 1 2 + 𝑦 2 − 𝑦 1 2 2.

3.

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c a 5. 6.

7. 𝒑= 𝟏 𝟒𝒂 = 𝟏 𝟒 𝟏 𝟔 = 𝟏 𝟐 𝟑

8.

9.

Conic Section – A figure formed by the intersection of a plane and a right circular cone

6.2 Equations of Circles +9+4 Completing the square when a=0 Circle with radius r & center (0,0) Completing the square when a=0 𝑥 2 +𝑏𝑥+ 𝑏 2 2 = 𝑥+ 𝑏 2 2 +9+4

The Standard Form of the Equation of the Ellipse The standard form of an ellipse centered at the origin with the major axis of length 2a along the x-axis and a minor axis of length 2b along the y-axis, is: The standard form of an ellipse centered at the origin with the major axis of length 2a along the y-axis and a minor axis of length 2b along the x-axis, is: vertex 𝑭 𝟏 (0,c) vertex 𝑭 𝟏 𝑭 𝟐 vertex 𝑭 𝟐 (0,-c) vertex 𝑐 2 = 𝑎 2 − 𝑏 2 𝑐 2 = 𝑎 2 − 𝑏 2

The Standard Form of the Equation of the Ellipse The standard form of an ellipse centered at any point (h, k) with the major axis of length 2a parallel to the y-axis and a minor axis of length 2b parallel to the x-axis, is: The standard form of an ellipse centered at any point (h, k) with the major axis of length 2a parallel to the x-axis and a minor axis of length 2b parallel to the y-axis, is: (h, k) (h, k) 𝑐 2 = 𝑎 2 − 𝑏 2 𝑐 2 = 𝑎 2 − 𝑏 2 Foci: Foci:

To find the foci use c2 = a2 + b2 The Standard Equation of a Hyperbola With Center (0, 0) and Foci on the x-axis Horizontal Hyperbola The equation of a hyperbola with the center (0, 0) and foci on the x-axis is: 𝐁 𝟏 (0, b) The length of the rectangle is 2a. The height of the rectangle is 2b. The vertices are (a, 0) and (-a, 0). The foci are (c, 0) and (-c, 0). The slopes of the asymptotes are (-c, 0) A1 A2 (c, 0) F1 (-a, 0) (a, 0) F2 𝐁 𝟐 (0, -b) To find the foci use c2 = a2 + b2 The equations of the asymptotes are 𝒚= 𝒃 𝒂 𝒙 and 𝒚=− 𝒃 𝒂 𝒙.

The equations of the asymptotes are 𝒚= 𝒂 𝒃 𝒙 and 𝒚=− 𝒂 𝒃 𝒙. The Standard Equation of a Hyperbola with Center (0, 0) and Foci on the y-axis Vertical Hyperbola The equation of a hyperbola with the center (0, 0) and foci on the y-axis is: F1(0, c) A1(0, a) The length of the rectangle is 2b. The height of the rectangle is 2a. The vertices are (0, a) and (0, -a). The foci are (0, c) and (0, -c). The slopes of the asymptotes are B1(-b, 0) B2(b, 0) To find the foci use c2 = a2 + b2 A2(0, -a) F2(0, -c) The equations of the asymptotes are 𝒚= 𝒂 𝒃 𝒙 and 𝒚=− 𝒂 𝒃 𝒙.

𝑐 2 = 𝑎 2 − 𝑏 2 𝑐 2 = 𝑎 2 − 𝑏 2 Foci: Foci:

Parabola Equation center at the origin (0,0) 𝒑= 𝟏 𝟒𝒂 𝒂= 𝟏 𝟒𝒑 𝒐𝒓 − 𝟏 𝟒𝒑 𝟒. 𝒙=− 𝟏 𝟒𝒑 𝒚 𝟐 𝟏. 𝒚= 𝟏 𝟒𝒑 𝒙 𝟐 𝟐. 𝒚=− 𝟏 𝟒𝒑 𝒙 𝟐 𝟑. 𝒙= 𝟏 𝟒𝒑 𝒚 𝟐 p p p p p (p, 0) p p)

Parabola Equation center at (h,k) 𝒚= 𝟏 𝟒𝒑 𝒙−𝒉 𝟐 +𝒌 𝒚=− 𝟏 𝟒𝒑 𝒙−𝒉 𝟐 +𝒌 𝒙= 𝟏 𝟒𝒑 𝒚−𝒌 𝟐 +𝒉 𝒙=− 𝟏 𝟒𝒑 𝒚−𝒌 𝟐 +𝒉