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Question

If the distances of P(x,y) from A(1,5) and B(5,1) are equal, then

A
2x=y
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B
3x=2y
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C
3x=y
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D
2x=3y
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Solution

The correct option is D 3x=2y
Let the points be P(x,y), A(1,5) and B(5,1)

Distance between two points is (x2x1)2+(y2y1)2

PA=(1x)2+(5y)2=x2+1+2x+25+y210y=x2+y2+2x10y+26

Therefore, PA2=x2+y2+2x10y+26

Similarly,

PB=(5x)2+(1y)2=25+x210x+1+y22y=x2+y210x2y+26

Therefore, PB2=x2+y210x2y+26

Since it is given that the distances are equal therefore PA2=PB2 that is

x2+y2+2x10y+26=x2+y210x2y+26
2x10y=10x2y
2x+10x=2y+10y
12x=8y
3x=2y

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