Conic Sections — Formula Sheet

y² = 4ax

y² = -4ax

x² = 4ay

x² = -4ay

SP = PM

x²/a² + y²/b² = 1 (a > b)

x²/b² + y²/a² = 1 (a > b)

e = √(1 - b²/a²)

Length of latus rectum = 2b²/a

Foci = (±ae, 0) or (0, ±ae)

yy₁ = 2a(x + x₁) for y² = 4ax

(xx₁)/a² + (yy₁)/b² = 1

y - y₁ = (-y₁/(2a))(x - x₁)

(a²x)/x₁ - (b²y)/y₁ = a² - b²

The type of conic depends on the value of e

if e < 1 → ellipse, e = 1 → parabola, e > 1 → hyperbola.

Major axis = 2a, Minor axis = 2b.

Standard equation

\(y^2 = 4ax\) — opens right, vertex at origin.

Other forms

\(y^2 = -4ax\) (left), \(x^2 = 4ay\) (up), \(x^2 = -4ay\) (down).

Condition for line y = mx + c to be tangent to ellipse: c² = a²m² + b².

Standard form

x²/a² + y²/b² = 1.

Parabola has e = 1.

Standard form

y² = 4ax.