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Question

Find the equation to the locus of a point which is always equidistant from the points whose coordinates are (a+b,ab) and (ab,a+b)

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Solution

Let the point be P(h,k)
A(a+b,ab) and B(ab,a+b)
PA=(h(a+b))2+(k(ab))2PB=(h(ab))2+(k(a+b))2
Given PA=PB
(h(a+b))2+(k(ab))2=(h(ab))2+(k(a+b))2(h(a+b))2+(k(ab))2=(h+(ab))2+(k(a+b))2h2+(a+b)22h(a+b)+k2+(ab)22k(ab)=h2+(ab)22h(ab)+k2+(a+b)22k(a+b)2ah2bh2ak+2kb=2ah+2hb2ak2kb4bh=4kbh=k
Replacing h by x and k by y
x=y
is the required equation of locus.

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