Conic Section Equations

Conic section polynomial

Ax^2 + By^2 + Cx + Dy + E = 0

if the center is (0,0)

the equation is x^2 + y^2 = r^2

if the center is (h,k)

the equation is (x-h)^2 + (y-k)^2 = r^2.

horizontal ellipse (a,0) and (-a,0), a^2 > b^2

x^2 / a^2 + y^2 / b^2 = 1

vertical ellipse (0,a) and (0,-a), a^2 > b^2

x^2 / b^2 + y^2 / a^2 = 1

horizontal ellipse (h,k), a^2 > b^2

(x-h)^2 / a^2 + (y-k)^2 / b^2 = 1

vertical ellipse (h,k), a^2>b^2

(x-h)^2 / b^2 + (y-k)^2 / a^2 = 1

Hyperbola (opens to the sides)

(x-h)^2 / a2 - (y-k)^2 / b^2 = 1

Hyperbola (opens up and down)

(y-k)^2 / a^2 - (x-h)^2 / b^2 = 1

Parabola

y = Ax^2 + Bx + C

If A < 0

the parabola opens downward and its vertex is the highest point on the parabola

If A > 0

then the parabola opens upward and its vertex is the lowest point on the parabola

ellipse

AB > 0

circle

A = B, A and B not equal to 0

parabola

A = 0 or B = 0, but not both

line

A = 0 and B = 0

hyperbola

AB < 0