# similitude of parabolas

Let us take two parabolas

 $y\;=\;ax^{2}\quad\mbox{and}\quad y\;=\;bx^{2}$

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

 $ax^{2}=mx,$

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

 $\vec{u}\;=\;\left(\!\begin{array}[]{c}\frac{m}{a}\\ \frac{m^{2}}{a}\end{array}\!\right)\;=\;\frac{m}{a}\left(\!\begin{array}[]{c}1% \\ m\end{array}\!\right)$

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

 $\vec{v}\;=\;\left(\!\begin{array}[]{c}\frac{m}{b}\\ \frac{m^{2}}{b}\end{array}\!\right)\;=\;\frac{m}{b}\left(\!\begin{array}[]{c}1% \\ m\end{array}\!\right)$

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

 $a\vec{u}\;=\;b\vec{v}$

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 SimilitudeOfParabolas 2013-03-22 18:51:04 2013-03-22 18:51:04 pahio (2872) pahio (2872) 8 pahio (2872) Theorem msc 51N20 msc 51N10 similarity of parabolas Homothety