If P(x, y) is a point equidistant from the points A(6, – 1) and B(2, 3), show that × – y = 3.

Question

Philoid · Accepted Answer

By distance formula, as shown below:

PA = √{(6 – x)² + (– 1 – y)²}

= √{(36 + x² –12x) + (1 + y² + 2y)}

⇒ PA = √{37 + x² – 12x + y² + 2y}

PB = √{(2 – x)² + (3 – y)²}

= √{(4 + x² – 4x + 9 + y² – 6y)}

⇒ PB = √{(13 + x² – 4x + y² – 6y)}

Given: PA = PB

Squaring both sides, we get

(37 + x² – 12x + y² + 2y) = (13 + x² – 4x + y² – 6y)

24 = 8x – 8y

Dividing by 8

x – y = 3

Hence proved.