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Question

# If the point A is symmetric to the point B≡(4,−1) with respect to the bisector of the first quadrant, then the length of AB is

A
5
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B
52
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C
32
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D
3
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Solution

## The correct option is B 5√2Let A≡(a,b).The coordinates of the mid point C of AB are (a+42,b−12).Since it lies on the line y=x (the bisector of first quadrant)∴b−12=a+42⇒a−b=−5Since AB is ⊥ to the line y=x, ...(1)∴ product of slope of AB and slope of the line y=x is equal to −1 ⇒b+1a−4×1=−1⇒b+1=−a+4⇒a+b=3Solving, we get a=−1,b=4.∴ Coordinates of the point A are (−1,4),∴AB=√(−1−4)2+(4+1)2=5√2

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