Derive the equation of a parabola from a focus and directrix.
Derive a horizontal parabola equation from focus and directrix
Problem
Derive the equation of the parabola with focus \((3,~1)\) and directrix \(x~=~-1\). Find the vertex midpoint and signed \(p\), identify the horizontal standard form, and substitute into \((y~-~k)^{2}~=~4p(x~-~h)\).
A vertical directrix forces a horizontal parabola, so the y-expression will be squared. We’ll project the focus horizontally, use the midpoint for the vertex and direction for signed p, equate focus distance with horizontal distance to the line, simplify after squaring, and verify the recovered focus and directrix.
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