8.6 Conic Sections Write equations of conic sections in standard form Identify conic sections from their equations

8.6 Conic Sections The equation of any conic section can be written in the general quadratic equation: Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 where A, B, and C ≠ 0 If you are given an equation in this general form, you can complete the square to write the equation in one of the standard forms you have already learned.

Standard Forms (you already know ) Conic Section Standard Form of Equation Parabolay = a(x – h) 2 + k x = a(y – k) 2 + h Circle(x – h) 2 + (y – k) 2 = r 2 Ellipse Hyperbola

Identifying Conic Sections Relationship of A and CType of Conic Section Only x 2 or y 2 Parabola Same number in front of x 2 and y 2 Circle Different number in front of x 2 and y 2 with plus sign Ellipse Different number in front of x 2 and y 2 with plus sign or minus sign Hyperbola

Example One: Write each equation in standard form. Then state whether the graph of the equation is a parabola, circle, ellipse, and hyperbola. y = x 2 + 4x + 1 x 2 + y 2 = 4x + 2 y 2 – 2x 2 – 16 = 0 x 2 + 4y 2 + 2x – 24y + 33 = 0

Example Two: Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, and hyperbola. x 2 + 2y 2 + 6x – 20y + 53 = 0 x 2 + y 2 – 4x – 14y + 29 = 0 3y 2 + x – 24y + 46 = 0 6x 2 – 5y x + 20y – 56 = 0

Your Turn: Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, and hyperbola. x 2 + y 2 – 6x + 4y + 3 = 0 6x 2 – 60x – y = 0 x 2 – 4y 2 – 16x + 24y – 36 = 0 x 2 + 2y 2 + 8x + 4y + 2 = 0