Question

If the ratio of the distance from P to A (5, -1) and B (7, 6) is 2:3 then in the locus of P, 3AP = ___​

Answer 1

Given info : The ratio of the distance from P to A(5, -1) and B(7, 6) is 2 : 3.

To find : The locus of P is ..

solution : let P(x , y)

given, PA/PB = 2/3

⇒3PA = 2PB

using distance formula,

PA = √{(x - 5)² + (y + 1)²}

PB = √{(x - 7)² + (y - 6)²}

now, 3√{(x - 5)² + (y + 1)²} = 2√{(x - 7)² + (y - 6)²}

squaring both sides

⇒9[(x - 5)² + (y + 1)² ] = 4[(x - 7)² + (y - 6)²]

⇒9x² + 9y² + 225 + 9 - 90x + 18y = 4x² + 4y² + 196 + 144 - 56x - 48y

⇒5x² + 5y² + (-90x + 56x) + (18y + 48y) + 234 - 340 = 0

⇒5x² + 5y² - 34x + 66y - 106 = 0

Therefore the locus of point P is 5x² + 5y² - 34x + 66y - 106 = 0.

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Answer 2

$\mathbb{ \ \ A\ N\ S\ W\ E\ R\: \: }$

If the ratio of the distance from P to A (5, -1) and B (7, 6) is 2:3 then in the locus of P, 3AP = 0