Two Dimensional Analytical Geometry-II — Formula Sheet

Standard form

(x - h)² + (y - k)² = r² where centre is (h,k) and radius is r

General form

x² + y² + 2gx + 2fy + c = 0 where centre is (-g,-f) and radius is √(g² + f² - c)

If e = 1: parabola; if 0 < e < 1: ellipse; if e > 1: hyperbola

General equation

Ax² + Bxy + Cy² + Dx + Ey + F = 0

Standard form

y² = 4ax (opens right); y² = -4ax (opens left); x² = 4ay (opens up); x² = -4ay (opens down)

For y² = 4ax: vertex (0,0), focus (a,0), directrix x = -a, axis = x-axis, latus rectum length = 4a

Vertex form with vertex at (h,k)

(y-k)² = 4a(x-h) or (x-h)² = 4a(y-k)

Standard form

x²/a² + y²/b² = 1 where a > b (major axis along x); or x²/b² + y²/a² = 1 where a > b (major axis along y)

For major axis along x

centre (0,0), vertices (±a,0), foci (±c,0) where c² = a² - b², eccentricity e = c/a < 1

Sum of focal distances constant

SP + S'P = 2a for any point P on ellipse

Standard form

(x-h)² + (y-k)² = r² with center (h,k) and radius r.

A parabola (e=1) is the locus where distance to focus equals distance to directrix.