Question

If A(4, 1) and B(5, 4), find the equation of the locus of point P if PA² = 3PB².

Answer 1

Let P(x, y) be any point on the required locus.

Given, A(4, 1), B(5, 4) and

PA² = 3PB²

∴ (x – 4)² + (y – 1)² = 3[(x – 5)² + (y – 4)²]

∴ x² – 8x + 16 + y² – 2y + 1 = 3(x² – 10x + 25 + y² – 8y + 16)

∴ x² – 8x + y² – 2y + 17 = 3x² – 30x + 75 + 3y² – 24y + 48

∴ 2x² + 2y² – 22x – 22y + 106 = 0

∴ x² + y² – 11x – 11y + 53 = 0

∴ The required equation of locus is

x² + y² – 11x – 11y + 53 = 0.