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[parent] tangent of conic section (Definition)

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

$\displaystyle Ax^2+By^2 +2Cxy+2Dx+2Ey+F = 0,$
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

$\displaystyle Ax_0x+By_0y+C(y_0x+x_0y)+D(x+x_0)+E(y+y_0)+F = 0.$
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

...#tex2html_wrap_inline203#1
This is true also in any skew-angled coordinate system.


"tangent of conic section" is owned by pahio.
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See Also: tangent line, tangent of circle, tangent plane of quadratic surface, quadratic inequality, conjugate diameters of ellipse, conjugate hyperbola, quadratic curves, equation of tangent of circle

Other names:  tangent of quadratic curve
Also defines:  polarising, polarizing, polarize, mixed term

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Cross-references: curve, point, parabola, hyperbola, ellipse, circle, tangent line, coordinate, parallel, skew-angled coordinate, conic section, equation
There are 5 references to this entry.

This is version 13 of tangent of conic section, born on 2004-07-16, modified 2008-06-15.
Object id is 6008, canonical name is TangentOfConicSection.
Accessed 6752 times total.

Classification:
AMS MSC: 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)
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