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.