If A(2, 0) and B(0, 3) are two points, find the equation of the locus

1.	If A(2, 0) and B(0, 3) are two points, find the equation of the locus of point P such that AP = 2BP.
Answer» Let P(x, y) be any point on the required locus. Given, A(2, 0), B(0, 3) and AP = 2BP ∴ AP² = 4BP² ∴ (x – 2)² + (y – 0)² = 4[(x – 0)² + (y – 3)²] ∴ x² – 4x + 4 + y² = 4(x² + y² – 6y + 9) x² – 4x + 4 + y² = 4x² + 4y² – 24y + 36 ∴ 3x² + 3 y² + 4x – 24y + 32 = 0 ∴ The required equation of locus is 3x² + 3y² + 4x – 24y + 32 = 0.

If A(2, 0) and B(0, 3) are two points, find the equation of the locus of point P such that AP = 2BP.

Answer»

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

Given, A(2, 0), B(0, 3) and AP = 2BP

∴ AP² = 4BP²

∴ (x – 2)² + (y – 0)² = 4[(x – 0)² + (y – 3)²]

∴ x² – 4x + 4 + y² = 4(x² + y² – 6y + 9) x² – 4x + 4 + y² = 4x² + 4y² – 24y + 36

∴ 3x² + 3 y² + 4x – 24y + 32 = 0

∴ The required equation of locus is 3x² + 3y² + 4x – 24y + 32 = 0.