Legend: ^ indicates an exponent (2^2 is 2 squared or 4) * indicates multiplication (2*2 is 4) sqrt() indicates a square root (sqrt(4) is 2) 1. List of Conic Sections Circle Ellipse Parabola Hyperbola 2. Features of Conic Sections A. Circles Equation: x^2 + y^2 = a^2 Eccentricity: 0 Linear Eccentricity: 0 Semi-Latus Rectum: a Focal Parameter: Infinite B. Ellipses Equation: (x^2/a^2) + (y^2/b^2) = 1 Eccentricity: sqrt(1 - (b^2/a^2)) Linear Eccentricity: sqrt(a^2 - b^2) Semi-Latus Rectum: b^2 / a Focal Parameter: b^2 / sqrt(a^2 - b^2) C. Parabolas Equation: y^2 = 4ax Eccentricity: 1 Linear Eccentricity: a Semi-Latus Rectum: 2a Focal Parameter: 2a D. Hyperbolas Equation: (x^2 / a^2) - (y^2 / b^2) = 1 Eccentricity: sqrt(1 + (b^2 / a^2)) Linear Eccentricity: sqrt(a^2 + b^2) Semi-Latus Rectum: b^2 / a Focal Parameter: b^2 / sqrt(a^2 + b^2) E. Relations of Features Focal Parameter * Eccentricity = Semi-Latus Rectum a * Eccentricity = Linear Eccentricity 3. Standard Form Equation of a Conic Section A. The Equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 Where A, B, and C are not all 0 B. Determining the type of Conic Section Using the Equation B^2 - 4AC we can determine the type of a non-degenerate conic section. If B^2 - 4AC is less than 0 the equation is an Ellipse. If B^2 - 4AC is less than 0 and A = C and B = 0 the equation is a Circle If B^2 - 4AC is equal to 0 then the equation is a Parabola If B^2 - 4AC is greater than 0 the equation is a Hyperbola If B^2 - 4AC is greater than 0 and A + C = 0 the equation is a Rectangular Hyperbola 4. Step by Step In this section I'm going to run through what the equation of the given type of Conic Section in both the standard for variables and using variables that use readable names. A. Circles Variables: Center Point (h, k), Radius r Equations: (x - h)^2 + (y - k)^2 = r^2 Plaintext: (x - The Center X Value)^2 + (y - The Center Y Value)^2 = Radius^2 B. Ellipses If the Bigger Axis is Horizontal Variables: Center Point (h, k), Length of Big Axis 2a, Length of Small Axis 2b Equations: ((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1 Plaintext: ((x - Center X Value)^2 / Half of the Length of the Big Axis^2) + ((y - Center Y Value)^2 / Half of the Length of the Small Axis^2) = 1 If the Bigger Axis is Vertical Variables: Center Point (h, k), Length of Big Axis 2a, Length of Small Axis 2b Equations: ((x - h)^2 / b^2) + ((y - k)^2 / a^2) = 1 Plaintext: ((x - Center X Value)^2 / Half of the Length of the Small Axis^2) + ((y - Center Y Value)^2 / Half of the Length of the Big Axis^2) = 1 Notes: To find the distance between the center of the Ellipse and the Foci use c^2 = a^2 - b^2 where a > b > 0 C. Hyperbolas If The Transverse Axis is Horizontal Variables: Center Point (h, k), Distance between Vertices 2a, Distance between Foci 2c Equations: ((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1 Plaintext: ((x - Center X Value)^2 / Half of the Distance between Vertices^2) - ((y - Center Y Value)^2 / b^2) = 1 If The Transverse Axis is Vertical Variables: Center Point (h, k), Distance between Vertices 2a, Distance between Foci 2c Equations: ((y - k)^2 / a^2) - ((x - h)^2 / b^2) = 1 Plaintext: ((y - Center Y Value)^2 / Half of the Distance between Vertices^2) - ((x - Center X Value)^2 / b^2) = 1 Notes: To find b^2 use c^2 = a^2 + b^2 D. Parabolas If Axis is Horizontal and p DOES NOT EQUAL 0 Variables: Vertex (h, k), Focus (h + p, k), p Equations: (y - k)^2 = 4p(x - h) Plaintext: (y - Vertex Y Value)^2 = 4p(x - Vertex X Value) Notes: Directrix is the line x = h - p, Axis is the line y = k If Axis is Vertical and p DOES NOT EQUAL 0 Variables: Vertex (h, k), Focus (h + p, k), p Equations: (x - h)^2 = 4p(y - k) Plaintext: (x - Vertex X Value)^2 = 4p(Y - Vertex Y Value) Notes: Directrix is the line y = k - p, Axis is the line x = h 5. Sources http://hotmath.com/hotmath_help/topics/conic-sections-and-standard-forms-of-equations.html http://en.wikipedia.org/wiki/Conic_section