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Question

Find the coordinates of the point equidistant from three given points A(5, 3), B(5, -5) and C(1, -5).


Solution

Let the required point be P(x,y). Then AP=BP=CP

That is, (AP)2=(BP)2=(CP)2

This means, (AP)2=(BP)2

(x5)2+(y3)2=(x5)2+(y+5)2

x210x+25+y26y+9=x210x+25+y2+10y+25

x2+y210x6y+34=x2+y210x+10y+50

x2+y210x6yx2y2+10x10y=5034

16y=16

y=16/16=1

And (BP)2=(CP)2

(x5)2+(y+5)2=(x1)2+(y+5)2

x210x+25+y2+10y+25=x22x+1+y2+10y+25

x2+y210x+10y+50=x2+y22x+10y+26

x2+y210x+10yx2y2+2x10y=2650

8x=24

y=24/8=3

Hence the required point is (3,-1)

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