If A(4,1) and B(5,4), find the equation of the locus of point P such t

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

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

Answer»

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