tangent of conic section

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

$$A{x}^2+B{y}^2+2Cxy+2Dx+2Ey+F=0,$$

where $A$, $B$, $C$, $D$, $E$ and $F$ are constants and  $A^2+B^2+C^2>0.$   (The $2Cxy$ is present only if the axes are not parallel to the coordinate axes.)

The equation of the tangent line of an ordinary conic section (i.e., circle, ellipse, hyperbola and parabola) in the point $(x_0,y_0)$ of the curve is

$$A{x}_{0}x+B{y}_{0}y+C(y_{0}x+x_{0}y)+D(x+x_0)+E(y+y_0)+F=0.$$

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

$x^2$ with $x_{0}x$,  $y^2$ with $y_{0}y$,  $2xy$ with $y_{0}x+x_{0}y$,  $2x$ with $x+x_0$,  $2y$ with $y+y_0$.

Examples:  The of the ellipse  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$   is  $\frac{x_{0}x}{a^2}+\frac{y_{0}y}{b^2}=1$, the of the hyperbola  $xy=\frac{1}{2}$   is  $y_{0}x+x_{0}y=1$.

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