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 $A^2+B^2+C^2 > 0.$ ¹ (The mixed term $2Cxy$ is present only if the principal 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 $$Ax_0x+By_0y+C(y_0x+x_0y)+D(x+x_0)+E(y+y_0)+F = 0.$$ 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 $x_0x$ , $y^2$ with $y_0y$ , $2xy$ with $y_0x+x_0y$ , $2x$ with $x+x_0$ , $2y$ with $y+y_0$ .

Examples: The tangent of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$ is $\frac{x_0x}{a^2}+\frac{y_0y}{b^2} = 1$ , the tangent of the hyperbola $xy = \frac{1}{2}$ is $y_0x+x_0y = 1$ .

Footnotes

... $A^2+B^2+C^2 > 0.$¹

This is true also in any skew-angled coordinate system.