Question

If A(-2,2,3) and B(13,-3,13) are two points .find the locus of a point p which moves in such a way that 3PA=2PB

Answer 1

We have to find the locus of a point P which moves in such a way that 3PA = 2PB , where A (-2, 2, 3) and B(13, -3, 13)

Solution : let point P = (x , y , z)

distance between P and A, PA = √{(x + 2)² + (y - 2)² + (z - 3)²}

distance between P and B, PB = √{(x - 13)² + (y + 3)² + (z - 13)²}

a/c to question,

3PA = 2PB

⇒9PA² = 4PB²

⇒9[(x + 2)² + (y - 2)² + (z - 3)²] = 4[(x - 13)² + (y + 3)² + (z - 13)²]

⇒9(x + 2)² - 4(x - 13)² + 9(y - 2)² - 4(y + 3)² + 9(z - 3)² - 4(z - 13)² = 0

⇒5(x² + y² + z² + 28x - 12y + 10z - 247) = 0

⇒x² + y² + z² + 28x - 12y + 10z - 247 = 0

Therefore the locus of the point p which moves in such a way that 3PA = 2PB, is x² + y² + z² + 28x - 12y + 10z - 247 = 0