A family of lines is given by three functions a, b, c, defining the lines a(t)x+b(t)y=c(t). A cursor allows to draw the line for a given t, as well as the point on the associated [url=https://en.wikipedia.org/wiki/Envelope_(mathematics)]envelope[/url].

Show that the point [math]\gamma(t)[/math] is [math]M^{-1}\left(\begin{matrix}c\\c'\end{matrix}\right)[/math] where [math]M[/math] is the matrix [math]\left(\begin{matrix}a&b\\a'&b'\end{matrix}\right)[/math].