The Parabola

The Parabola

A parabola is the locus of a point that moves such that its distance from a fixed point (called the focus) is equal to its perpendicular distance from a fixed line (called the directrix).

Key Components

• Focus ($F$): A fixed point inside the curve.
• Directrix ($D$): A fixed line outside the curve.
• Vertex ($V$): The midpoint between the focus and the directrix. It is the turning point of the parabola.
• Axis of Symmetry: The line passing through the focus and perpendicular to the directrix.
• Latus Rectum ($LR$): The chord passing through the focus and perpendicular to the axis of symmetry. Its length is $4a$ (or $4p$).

Standard Equations

Let $(h, k)$ be the vertex and $a$ be the distance from the vertex to the focus (focal length). Note that $a > 0$.

Vertical Parabola (Opening Up or Down)

Vertical Parabola Equation

$(x - h)^2 = \pm 4a (y - k)$

• Opening Up: $+4a$. Focus $(h, k+a)$. Directrix $y = k-a$.
• Opening Down: $-4a$. Focus $(h, k-a)$. Directrix $y = k+a$.

Horizontal Parabola (Opening Right or Left)

Horizontal Parabola Equation

$(y - k)^2 = \pm 4a (x - h)$

• Opening Right: $+4a$. Focus $(h+a, k)$. Directrix $x = h-a$.
• Opening Left: $-4a$. Focus $(h-a, k)$. Directrix $x = h+a$.

General Equation

The general equation of a parabola is:

• Vertical: $Ax^2 + Dx + Ey + F = 0$ ($B=0, C=0$)
• Horizontal: $Cy^2 + Dx + Ey + F = 0$ ($A=0, B=0$)

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