similitude of parabolas

Two parabolasPlanetmathPlanetmath need not be congruent, but they are always similarMathworldPlanetmathPlanetmath.  Without the definition of parabola by focus and directrix, the fact turns out of the simplest equation  y=ax2  of parabola.

Let us take two parabolas


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


whence the abscissaMathworldPlanetmath of the other point of intersection is ma; the corresponding ordinate is thus m2a.  So, this point has the position vector


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


Accordingly, those position vectors have the depencence


for all values of the slope m of the cutting line.  This means that both parabolas are homotheticMathworldPlanetmath 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