Conic Sections

A conic section is the locus

A conic section is the locus of a point P that moves such that its distance from a fixed point (focus) bears a constant ratio (eccentricity e) to its

Standard form with vertex at origin

Standard form with vertex at origin, focus at (a,0), directrix x = -a, and latus rectum = 4a. Other forms: y² = -4ax (opens left), x² = 4ay (opens up)

Standard form with center at origin

Standard form with center at origin, major axis = 2a, minor axis = 2b, foci at (±ae,0) where e² = 1 - b²/a², and latus rectum = 2b²/a. Example: For x²

For parabola y² = 4ax at

For parabola y² = 4ax at point (x₁,y₁): Tangent: yy₁ = 2a(x + x₁), Normal: (y - y₁) = -(y₁/2a)(x - x₁). For ellipse x²/a² + y²/b² = 1 at (x₁,y₁): Tang

For external point (x₁

For external point (x₁,y₁), chord of contact with parabola y² = 4ax is yy₁ = 2a(x + x₁). Pole of line lx + my + n = 0 with respect to parabola is (-n/