If the point P(x, y) is equidistant from the points A(5, 1) and B(– 1, 5), prove that 3x = 2y.

The point P(x, y) is equidistant from the points A(5, 1) and B(– 1, 5), means PA = PB

By distance formula, as shown below:

PA = √{(5 – x)2 + (1 – y)2}

= √{(25 + x2 – 10x) + (1 + y2 – 2y)}

PA = √{26 + x2 – 10x + y2 – 2y}

PB = √{(– 1 – x)2 + (5 – y)2}

= √{(1 + x2 + 2x + 25 + y2 – 10y)}

PB = √{(26 + x2 + 2x + y2 – 10y)}

Now, PA = PB

Squaring both sides, we get

26 + x2 – 10x + y2 – 2y = 26 + x2 + 2x + y2 – 10y

12x = 8y

3x = 2y

Hence proved.

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