📂Geometry

Parabola

Definition ¹

For a point $F$ on the plane and a line $l$ that does not pass through it, the set of points equidistant from $F$ and $l$ is called a parabola.

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• $F$ is called the focus.
• $l$ is called the directrix.
• The line passing through $F$ and perpendicular to $l$ is called the axis of the parabola.
• The intersection of the axis and the parabola is called the vertex.

Description

If the distance between a point $P$ and the focus is $\left| PF \right|$, and the distance to the directrix $l$ is $\left| Pl \right|$, then a parabola with focus and directrix at $F, l$ is described by the following set. $$\left\{ P : \left| PF \right| = \left| Pl \right| \right\}$$ It is one of the conic sections.

The motion of an object launched from the ground at an angle of $\theta$ and a speed of $v_{0}$ is called parabolic motion.

Discriminant

For a given quadratic curve $Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0$, $\Delta = B^{2} - 4AC$ is called the discriminant. A quadratic curve with a discriminant of $0$ is a parabola.