Question

if A (4,1) B (5,4) find equation of locus of point P if PA^2=3PB^2​

Answer 1

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

PA² = (x-4)² + (y-1)²

PA² = x² + 16 - 8x + y² + 1 - 2y

PA² = x² + y² - 8x - 2y + 17

PB² = (x-5)² + (y-4)²

PB² = x² + 25 - 10x + y² + 16 - 8y

PB² = x² + y² - 10x - 8y + 41

3PB² = 3 ( x² + y² - 10x - 8y + 41)

3PB² = 3x² + 3y² - 30x - 24y + 123

Now, Given equation is PA² = 3PB²

⇒ x² + y² - 8x - 2y + 17 = 3x² + 3y² - 30x - 24y + 123

⇒ 3x² + 3y² - 30x - 24y + 123 - ( x² + y² - 8x - 2y + 17) = 0

⇒2x² + 2y² - 22x - 22y + 106 = 0

Therefore, The equation of the locus is 2x² + 2y² - 22x - 22y + 106 = 0