If the co-ordinates of the two points \( A \) and \( B \) are \( (3,4) \) and \( (5,-2) \) , respectively. The co-ordinates of any point P if \( \mathrm{PA}=\mathrm{PB} \) and area of \( \triangle \mathrm{PAB}=10 \) sq. units are.
PA = PB means P is on the perpendicular bisector of AB.
Midpoint of AB = (4, 1). Slope of AB = \(\frac{-2-4}{5-3} = -3\), so the bisector has slope \(\frac{1}{3}\): \(x - 3y = 1\).
Area = 10: with P(x, y), \(\frac{1}{2}|3(-2-y) + 5(y-4) + x(4+2)| = 10 \Rightarrow |6x + 2y - 26| = 20\)
So \(3x + y = 23\) or \(3x + y = 3\).
Answer: (A) (7, 2) or (1, 0)
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