# similitude of parabolas

Two parabolas^{} need not be congruent, but they are always similar^{}. Without the definition of parabola by focus and directrix, the fact turns out of the simplest equation $y=a{x}^{2}$ of parabola.

Let us take two parabolas

$$y=a{x}^{2}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}y=b{x}^{2}$$ |

which have the origin as common vertex and the $y$-axis as common axis. Cut the parabolas with the line $y=mx$ through the vertex. The first parabola gives

$$a{x}^{2}=mx,$$ |

whence the abscissa^{} of the other point of intersection is $\frac{m}{a}$; the corresponding ordinate is thus
$\frac{{m}^{2}}{a}$. So, this point has the position vector

$$\overrightarrow{u}=\left(\begin{array}{c}\hfill \frac{m}{a}\hfill \\ \hfill \frac{{m}^{2}}{a}\hfill \end{array}\right)=\frac{m}{a}\left(\begin{array}{c}\hfill 1\hfill \\ \hfill m\hfill \end{array}\right)$$ |

Similarly, the cutting point of the line and the second parabola has the position vector

$$\overrightarrow{v}=\left(\begin{array}{c}\hfill \frac{m}{b}\hfill \\ \hfill \frac{{m}^{2}}{b}\hfill \end{array}\right)=\frac{m}{b}\left(\begin{array}{c}\hfill 1\hfill \\ \hfill m\hfill \end{array}\right)$$ |

Accordingly, those position vectors have the http://planetmath.org/node/848linear depencence

$$a\overrightarrow{u}=b\overrightarrow{v}$$ |

for all values of the slope $m$ of the cutting line. This means that both parabolas are homothetic^{} with respect to the origin and therefore also similar.

Title | similitude of parabolas |
---|---|

Canonical name | SimilitudeOfParabolas |

Date of creation | 2013-03-22 18:51:04 |

Last modified on | 2013-03-22 18:51:04 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 51N20 |

Classification | msc 51N10 |

Synonym | similarity of parabolas |

Related topic | Homothety |