SOLUTION: Use the definition of a parabola to show that the parabola with the vertex(h,k) and focus (h,k-c) has the equation {{{ (x-h)^2=-4c(y-k)}}}. Show all work.

Algebra ->  Algebra  -> Quadratic-relations-and-conic-sections -> SOLUTION: Use the definition of a parabola to show that the parabola with the vertex(h,k) and focus (h,k-c) has the equation {{{ (x-h)^2=-4c(y-k)}}}. Show all work.       Log On

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Question 83414: Use the definition of a parabola to show that the parabola with the vertex(h,k) and focus (h,k-c) has the equation +%28x-h%29%5E2=-4c%28y-k%29. Show all work.

Answer by scott8148(5896) About Me  (Show Source):
You can put this solution on YOUR website!
a parabola is the locus of points that are equidistant from a given point (focus) and a given line (directrix)

the vertex is located midway between the focus and the directrix, which means the equation for the directrix is y=k+c

using the distance formula, the equation for a point (x,y) is (x-h)^2+(y-(k-c))^2=(y-(k+c))^2

expanding gives (x-h)^2+y^2-2ky+2cy+k^2-2kc+c^2=y^2-2ky-2cy+k^2+2kc+c^2

subtracting y^2-2ky+k^2+c^2 gives (x-h)^2+2cy-2kc=-2cy+2kc ... adding 2kc-2cy gives (x-h)^2=4kc-4cy

factoring gives (x-h)^2=-4c(y-k)