## You are here

Hometangent of conic section

## Primary tabs

# 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 $A^{2}+B^{2}+C^{2}>0.$^{1}^{1}This is true also in any skew-angled coordinate system. (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

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

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 tangent 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 tangent of the hyperbola $xy=\frac{1}{2}$ is $y_{0}x+x_{0}y=1$.

## Mathematics Subject Classification

51N20*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections