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Distance Formula

If A -2,2,3 a...

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.

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Solution

Let P be (x, y, z), here, A(-2, 2, 3) and B(13, -3, 13) and 3PA = 2PB

$\Rightarrow 3 \sqrt{(- 2 - x)^{2} + (2 - y)^{2} + (3 - z)^{2}}$

= $2 \sqrt{(13 - x)^{2} + (- 3 - y)^{2} + (13 - z)^{2}}$

$\Rightarrow 9 [x^{2} + 4 x + y^{2} + 4 - 4 y + z^{2} + 9 - 6 z]$

= $4 [x^{2} + 169 - 26 x + y^{2} + 9 + 6 y + z^{2} + 169 - 26 z]$

$\Rightarrow 9 x^{2} - 4 x^{2} + 36 x + 104 x + 36 - 676 + 9 y^{2} - 4 y^{2} + 36 - 36 - 36 y - 24 y + 9 z^{2} - 4 z^{2} + 81 - 676 - 54 z + 6 y z = 0$

$\Rightarrow 5 x^{2} + 5 y^{2} + 6 z^{2} + 140 x - 60 y + 50 z - 1235 = 0$ is the locus of the point P

$\Rightarrow 5 (x^{2} + y^{2} + z^{2}) + 140 x - 60 y + 50 z - 1235 = 0$

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