1. 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,−
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

2. 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=−
𝑥 𝑦−7 2 =34
Center = (−5,7)
Radius = 34

3. 𝐷𝑖𝑠𝑡= 𝑥 2 − 𝑥 𝑦 2 − 𝑦 1 2 2.

4. 3.

5. 4.

6. c a 5. 6.

7. 7.

8. 8.

9. 9.

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

11. 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 = 𝑥+ 𝑏 2 2
+9+4

12. 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

13. 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:

14. 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.

15. 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.

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

17. 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)

18. 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