Derive the equation of a parabola from a focus and directrix.
Derive a vertical parabola equation from focus and directrix
Problem
Derive the equation of the parabola with focus \((0,~2)\) and directrix \(y~=~-2\). Find the vertex midpoint and signed \(p\), identify the vertical standard form, and substitute into \((x~-~h)^{2}~=~4p(y~-~k)\).
The equation can be derived directly from the parabola’s equal-distance definition. We’ll first locate the vertex and signed focal distance, set a general point’s focus distance equal to its perpendicular distance from the horizontal directrix, square the nonnegative distances, cancel matching terms, and compare with vertical standard form.
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