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Question

# Let A=(1,0) , B=(−1,0),C=(2,0), the locus of a point P such that PB2+PC2=2PA2 is

A
a straight line parallel to x-axis
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B
a straight line parallel to y-axis
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C
pair of straight line
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D
combined equation of coordiante axes
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Solution

## The correct option is B a straight line parallel to y-axisLet P=(x,y)Then, PA=√(x−1)2+y2 ...(i)PB=√(x+1)2+y2 ...(ii)PC=√(x−2)2+y2 ...(iii)Hence, PC2+PB2=2PA2⇒[(x−2)2+y2]+[(x+1)2+y2]=2(x−1)2+2y2⇒(x−2)2+(x+1)2+2y2=2(x−1)2+2y2⇒(x−2)2+(x+1)2=2(x−1)2⇒(x2−4x+4)+(x2+2x+1)=2(x2−2x+1)⇒2x2−2x+5=2x2−4x+2⇒2x+3=0⇒x=−32Hence, this represents a line parallel to y-axis.

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