Parabolas

A parabola is a fundamental curve in the family of conic sections arising from the intersection of a plane with a double-napped cone when the plane is parallel to the cone’s slant height. This geometric condition yields a unique open curve defined by its equidistance from a fixed point, the focus, and a fixed line, the directrix.

A parabola is mathematically defined as the locus of all points in a plane that are equidistant from the focus and the directrix. In Cartesian coordinates, the standard form depends on the orientation of the axes:

For a vertically oriented parabola, the standard form is:

For a horizontally oriented parabola, the standard form is:

Here, (h, k) denotes the vertex of the parabola, and p is the distance from the vertex to the focus. If the vertex is at the origin, the equation simplifies to y² = 4px and x² = 4py, respectively. The sign of p determines whether the parabola opens in the positive or negative direction along the axis.

The axis of symmetry of a parabola is a central line that passes through the vertex and focus, perpendicular to the directrix. The position of the focus relative to the vertex determines the opening direction: upward or rightward if the focus lies in the positive axis direction and downward or leftward if in the negative. The vertex, the point equidistant from both the focus and directrix, is a pivotal reference in defining the curve's symmetry and orientation.

Parabolas exhibit a significant reflective property: the tangent at any point on the parabola makes equal angles with the line to the focus and the axis, resulting from the law of reflection and the curve's geometric symmetry.