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Question

Find the equation to the parabola with focus (a, b) and directrix xa+yb=1.

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Solution

Let P(x,y) be a point on the parabola with the directrix M:xa+yb=1 and focus S:(a,b)
we know that ,SP2=PM2
(xa)2+(yb)2 =(xa+yb1)1a2+1b22
After rearranging the terms in both sides,we get
a2x2+b2y2+a2y2+b2x22a3x2yba22axb22yb3+a4+2a2b2+b4 =
a2y2+b2x2+a2b2+2abxy2b2xa2a2yb
After simplifying both sides,we get
(axby)22a3x2b3y+a4+a2b2+b4=0

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