Parabolas part 1

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A Parabola is the set of all points (locus) equal distance from a fixed line (Directrix) and a fixed point (Focus).  The Vertex is equal distance from the directrix and the focus. The equation of a parabola is derived from the distance formula. The standard equation of a parabola with vertex at the origin and directrix horizontal is

X^2 = 4pY

solving for Y

y=\frac{x^2}{4p}

p>0  the curve opens up

p<0  the curve opens down

 

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A parabola has the useful feature that a line parallel to the symmetry axis striking the parabola at any point is reflected onto the focus.

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recasting our most common parabola

y=x^2

y=\frac{x^2}{4p}

becomes

y=\frac{x^2}{4*0.25}

the focus is at (0, 0.25)

the directrix is at Y=-0.25

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