# 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

 $Ax^{2}+By^{2}+2Cxy+2Dx+2Ey+F=0,$

where $A$, $B$, $C$, $D$, $E$ and $F$ are constants and  $A^{2}+B^{2}+C^{2}>0.$11This is true also in any skew-angled coordinate system.   (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

 $Ax_{0}x+By_{0}y+C(y_{0}x+x_{0}y)+D(x+x_{0})+E(y+y_{0})+F=0.$

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  $y_{0}x+x_{0}y=1$.