Conic Sections - Parabola

A parabola is defined as the set of all points in a plane that are equidistant from a fixed line called the directrix and a fixed point called the focus (which is not on the line). Visually, this creates a U-shaped curve where the distance from any point on the curve to the focus is exactly equal to the perpendicular distance to the directrix.

The Axis of Symmetry is the line passing through the focus and perpendicular to the directrix. The parabola is perfectly balanced on either side of this line, meaning if you fold the graph along this axis, the two halves of the curve will coincide.

The Vertex is the point where the parabola intersects its axis of symmetry. In standard form, the vertex is at the origin $(0, 0)$. It represents the 'turning point' of the curve and is exactly halfway between the focus and the directrix.

The Latus Rectum is a line segment perpendicular to the axis of symmetry, passing through the focus, with its endpoints lying on the parabola. Visually, it represents the width of the parabola at the focus. Its length is always $4a$, where $a$ is the distance from the vertex to the focus.

The standard forms of a parabola depend on its orientation: horizontal parabolas open to the right ($y^2 = 4ax$) or left ($y^2 = -4ax$), while vertical parabolas open upwards ($x^2 = 4ay$) or downwards ($x^2 = -4ay$). The direction is determined by which variable is squared and the sign of the constant term.

The Focal Distance of any point $P(x, y)$ on the parabola is the distance from the focus to that point. By definition, this is equal to the distance from the point $P$ to the directrix. For $y^2 = 4ax$, the focal distance of point $(x, y)$ is $|x + a|$.