The Parabola

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Parabola is a conic section. It is defined as the set of all points whose distance from a fixed point, called the focus, is equal to the distance from a fixed line, called the directrix.

The eccentricity, e of a parabola is equal to 1.

Focus (F) \(a\) distance from the vertex.
Vertex (v) \(h,k\)
Latus Rectum (LR) \(LR~=~4a\)
A line segment passing through the focus and parallel to the directrix.
\(a\) distance between the vertex and the focus or the distance between the directix and the vertex.

General Equation of Parabola

\( Ax^2 + Dx + Ey + F = 0\) or \(Cy^2 +Dx + Ey + F = 0\)

Where A, B, C, D E and F are constants.

Standard Equations of Parabola

Vertex at \( (0,0) \)
\(y^2 = 4ax\) Opens Upwards
\(y^2 = -4ax\) Opens Downwards
\(x^2 = 4ax\) Opens to the right
\(x^2 = -4ax\) Opens to the left
Vertex at \( (h,k) \)
\( (y-h)^2 = 4a(x-k)\) Opens Upwards
\((y-h)^2 = -4a(x-k)\) Opens Downwards
\( (x-k)^2 = 4a(y-h)\) Opens to the right
\((x-k)^2 = -4a(y-h)\) Opens to the left

How to solve \(h,k\) and \(a \) if the given equation is in its general form.

  • Transform to standard form and get the \(h,k\) values.
  • If the general form is: \( Ax^2 + Dx + Ey + F = 0\)
    • \(h = \frac{-D}{2A}\)
    • \(k = \frac{D^2-4AF}{4AE}\)
    • \(a = \frac{-E}{4A}\)
  • If the general form is: \(Cy^2 +Dx + Ey + F = 0\)
    • \(h = \frac{E^2-4CF}{4CD}\)
    • \(k = \frac{-E}{2C}\)
    • \(a = \frac{-D}{4C}\)