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Conic Section Formula SheetGeneral Formulas
Midpoint Distance Slope Slope-Intercept Point-slopex1 + x22
,y1 + y22
⎛⎝⎜
⎞⎠⎟
x2 − x1( )2 + y2 − y1( )2 y2 − y1x2 − x1
y = mx + b y − y1 = m x − x1( )
ParabolaVertical: Opens up or down
Equation:
Vertex: (h,k)Focus: (h, k + p)Directrix: y = k - pAxis of Symetry: x = hIf p > 0 parabola opens up, if p < 0 parabola opens down
x − h( )2 = 4p y − k( )Horizontal: Opens right or left
Equation:
Vertex: (h,k)Focus: (h + p, k)Directrix: x = h - pAxis of Symetry: y = kIf p > 0 parabola opens right, if p < 0 parabola opens left
Ellipse
y − k( )2 = 4p x − h( )
Horizontal: Major Axis Parallel to x - axis
Equation:
Center: (h,k)Vertices: (h ± a, k)Co-vertices: (h, k ± b)Foci: (h ± c, k); where c2 = a2 - b2
x − h( )2a2
+y − k( )2b2
= 1
Vertical: Major Axis Parallel to y - axis
Equation:
Center: (h,k)Vertices: (h, k ± a)Co-vertices: (h ± b, k)Foci: (h, k ± c); where c2 = a2 - b2
x − h( )2b2
+y − k( )2a2
= 1
HyperbolaHorizontal: Transverse Axis Parallel to x - axis
Equation:
Center: (h,k)Vertices: (h ± a, k)Foci: (h ± c, k); where c2 = a2 + b2
Asymptotes: y = ± bax − h( )+ k
Vertical: Transverse Axis Parallel to y - axis
Equation:
Center: (h,k)Vertices: (h, k ± a)Foci: (h, k ± c); where c2 = a2 + b2
Asymptotes: y = ± abx − h( )+ k
x − h( )2a2
−y − k( )2b2
= 1y − k( )2a2
−x − h( )2b2
= 1
CircleEquation: Center: (h,k)
Radius: r(x − h)2 + ( y − k)2 = r 2
Indentifying Conic SectionsTo identify conic sections of the equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 where A, B, and C do not all equal zero:
Circle: If B2 - 4AC < 0, with B = 0 and A = CEllipse: If B2 - 4AC < 0, with B ≠ 0 or A ≠ CHyperbola: If B2 - 4AC > 0Parabola: If B2 - 4AC = 0
