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=1 Circle with radius r & center (0,0) Completing the square when a=1 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