# 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  $y_{0}x+x_{0}y=1$.