Polynomial stories

Polynomials beneath the antennas -------------------------------

How do we choose the shape of antennas designed to reflect incoming rays and focus them as effectively as possible? The parabola with focus F and directrix d (at a distance p, the parameter, from F) is the set of points M equidistant from d and F. Its equation is $y={\displaystyle \frac{x^2}{2p}}y\; =\; \backslash dfrac\{x^2\}\{2p\}$ in the orthonormal coordinate system with its vertex S as origin and its tangent at the vertex as the x-axis.

At a point M on this curve with x-coordinate a, the tangent line has equation $y={\displaystyle \frac{a}{p}}x-{\displaystyle \frac{a^2}{2p}}\mathrm{.}y\; =\; \backslash dfrac\{a\}\{p\}\; x\; -\; \backslash dfrac\{a^2\}\{2p\}.$

It therefore intersects the parabola's axis (SF) at $\text{T}(0,-{\displaystyle \frac{a^2}{2p}})\mathrm{.}\backslash text\{T\}\; \backslash left(\; 0,\; -\backslash dfrac\{a^2\}\{2p\}\; \backslash right).$