Tangents and the parabola: gems galore

Along with the circle and the ellipse, the parabola is one of the simplest plane curves there is. It can be described as a set of points satisfying an easily stated geometric property. Take a point F and a line Δ that does not pass through F. The set of points equidistant from F and Δ forms a parabola (see the article "A Line, a Point, That's All"). The point F is its focus, and the line Δ its directrix. Starting from this definition makes it possible to explore the properties of this curve geometrically. You may prefer a more analytical approach. By choosing a suitable orthonormal coordinate system, a parabola can be regarded as a curve with equation y = mx² where m is a strictly positive real number. In that case, the focus F has coordinates $(0;{\displaystyle \frac{1}{4m}})\backslash left(\; 0\backslash ;\; ;\; \backslash dfrac\{1\}\{4m\}\; \backslash right)$ and the directrix has equation $y=-{\displaystyle \frac{1}{4m}}\mathrm{.}y=\; -\; \backslash dfrac\{1\}\{4m\}.$

and directrix Δ has a simple equation: y = *mx ².

Tangents and chords ---------------------------