# equation of tangent of circle

We derive the equation of tangent line for a circle with radius $r$. For simplicity, we chose for the origin the centre of the circle, when the points $(x,y)$ of the circle satisfy the equation

${x}^{2}+{y}^{2}={r}^{2}.$ | (1) |

Let the point of tangency be $({x}_{0},{y}_{0})$. Then the slope of radius with end point^{} $({x}_{0},{y}_{0})$ is
$\frac{{y}_{0}}{{x}_{0}}$, whence, according to the parent entry (http://planetmath.org/TangentOfCircle), its opposite inverse
$-\frac{{x}_{0}}{{y}_{0}}$ is the slope of the tangent, being perpendicular^{} to the radius. Thus the equation of the tangent is written as

$$y-{y}_{0}=-\frac{{x}_{0}}{{y}_{0}}(x-{x}_{0}).$$ |

Removing the denominator and the parentheses we obtain from this first ${x}_{0}x+{y}_{0}y={x}_{0}^{2}+{y}_{0}^{2}$, and then

${x}_{0}x+{y}_{0}y={r}^{2}$ | (2) |

since $({x}_{0},{y}_{0})$ satisfies (1).

Remark. In the equation (2) of the tangent, ${x}_{0}$, ${y}_{0}$ are the coordinates^{} of the point of tangency and
$x,y$ the coordinates of an arbitrary point of the tangent line. But one can of course swap those meanings; then we interprete (2) such that ${x}_{0}$, ${y}_{0}$ are the coordinates of some point $P$ outside the circle (1) and $x,y$ the coordinates of the point of tangency of either of the tangents which may be drawn from $P$ to the circle. If (2) now is again interpreted as an equation of a line (its degree (http://planetmath.org/AlgebraicEquation) is 1!), this line must pass through both the mentioned points of tangency $A$ and $B$ (they satisfy the equation!); in a , (2) is now the equation of the tangent chord $AB$ of $P=({x}_{0},{y}_{0})$. See also http://mathworld.wolfram.com/Polar.htmlpolar.

Title | equation of tangent of circle |
---|---|

Canonical name | EquationOfTangentOfCircle |

Date of creation | 2013-03-22 18:32:16 |

Last modified on | 2013-03-22 18:32:16 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 10 |

Author | pahio (2872) |

Entry type | Derivation |

Classification | msc 51M20 |

Classification | msc 51M04 |

Related topic | Polarising |