# 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

$$A{x}^{2}+B{y}^{2}+2Cxy+2Dx+2Ey+F=0,$$ |

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 $2Cxy$ 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

$$A{x}_{0}x+B{y}_{0}y+C({y}_{0}x+{x}_{0}y)+D(x+{x}_{0})+E(y+{y}_{0})+F=0.$$ |

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 ${y}_{0}x+{x}_{0}y=1$.

Title | tangent of conic section |

Canonical name | TangentOfConicSection |

Date of creation | 2013-03-22 14:28:40 |

Last modified on | 2013-03-22 14:28:40 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 16 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 51N20 |

Synonym | tangent of quadratic curve |

Related topic | TangentLine |

Related topic | TangentOfCircle |

Related topic | TangentPlaneOfQuadraticSurface |

Related topic | QuadraticInequality |

Related topic | ConjugateDiametersOfEllipse |

Related topic | ConjugateHyperbola |

Related topic | QuadraticCurves |

Related topic | EquationOfTangentOfCircle |

Related topic | TangentOfHyperbola |

Defines | polarising |

Defines | polarizing |

Defines | polarize |

Defines | mixed term |