tangent of conic section

The equation of every conic sectionMathworldPlanetmath (and the degenerate cases) in the rectangular (x,y)-coordinate systemMathworldPlanetmath may be written in the form


where A, B, C, D, E and F are constants and  A2+B2+C2>0.11This is true also in any skew-angled coordinate system.   (The 2Cxy is present only if the axes are not parallelMathworldPlanetmathPlanetmath to the coordinate axes.)

The equation of the tangent line of an ordinary conic section (i.e., circle, ellipseMathworldPlanetmath, hyperbolaMathworldPlanetmath and parabola) in the point (x0,y0) of the curve is


Thus, the equation of the tangent line can be obtained from the equation of the curve by polarizing it, i.e. by replacing

x2 with x0x,  y2 with y0y,  2xy with y0x+x0y,  2x with x+x0,  2y with y+y0.

Examples:  The of the ellipse  x2a2+y2b2=1   is  x0xa2+y0yb2=1, the of the hyperbola  xy=12   is  y0x+x0y=1.

Title tangent of conic section
Canonical name TangentOfConicSection
Date of creation 2013-03-22 14:28:40
Last modified on 2013-03-22 14:28:40
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 16
Author pahio (2872)
Entry type Definition
Classification msc 51N20
Synonym tangent of quadratic curve
Related topic TangentLine
Related topic TangentOfCircle
Related topic TangentPlaneOfQuadraticSurface
Related topic QuadraticInequality
Related topic ConjugateDiametersOfEllipse
Related topic ConjugateHyperbola
Related topic QuadraticCurves
Related topic EquationOfTangentOfCircle
Related topic TangentOfHyperbola
Defines polarising
Defines polarizing
Defines polarize
Defines mixed term