unit 9 part 1 formulas

circles, ellipses, parabolas, hyperbolas

circle (standard form)

(x-h)^2 + (y-k)^2 = r^2

center at (h,k)

r = radius

VERTICAL-opening parabola (standard form)

(x-h)^2 = 4p(y-k)

vertex: (h,k)

p: distance between focus/directrix and vertex

HORIZONTAL-opening parabola (standard form)

(y-k)^2 = 4p(x-h)

vertex: (h,k)

p: distance between focus/directrix and vertex

tangent line of a parabola

y = mx - yb

yb = the y-coordinate of point b

point b = (focus - d)

d = sqrt((x2 - x1)^2 + (y2 - y1)^2) (use distance formula)

m = slope ((y2 - y1) / (x2 - x1)) where one of the points is b

LONG ellipse (standard form)

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

center: (h,k)

a: major axis

b: minor axis

c^2 = a^2 - b^2

vertices: (h + a, k) (h - a, k) (h, k + b) (h, k - b)

foci: (h + c, k) (h - c, k)

TALL ellipse (standard form)

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

center: (h,k)

vertices: (h + b, k) (h - b, k) (h, k + a) (h, k - a)

foci: (h, k + c) (h, k - c)

eccentricity

e = c/a

hyperbola (standard form) (t.a.v.)

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

center: (h,k)

A: distance between a vertex and the center

C: distance between a focus and the center

b^2 = c^2 - a^2

hyperbola (standard form) (t.a.h.)

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

center: (h,k)

hyperbolic asymptotes (t.a.h.)

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

y = k - (b/a)(x - h)

hyperbolic asymptotes (t.a.v.)

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

y = k - (a/b)(x - h)

circle discriminant

b^2 - 4ac < 0 (a = c)

parabola discriminant

b^2 - 4ac = 0

ellipse discriminant

b^2 - 4ac < 0 (a and c have same signs)

hyperbola discriminant

b^2 - 4ac > 0