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

Given: Points are A(-2, 2, 3) and B(13, -3, 13)


To find: the locus of point P which moves in such a way that 3PA = 2PB


Let the required point P(x, y, z)


According to the question:


3PA = 2PB


⇒ 9PA2 = 4PB2


Formula used:


The distance between any two points (a, b, c) and (m, n, o) is given by,



Therefore,


The distance between P(x, y, z) and A(-2, 2, 3) is PA,




The distance between P(x, y, z) and B(13, -3, 13) is PB,




As 9PA2 = 4PB2


9{(x + 2)2+ (y – 2)2 + (z – 3)2} = 4{(x – 13)2 + (y + 3)2 + (z – 13)2}


⇒ 9{x2+ 4 + 4x + y2 + 4 – 4y + z2 + 9 – 6z} = 4{x2+ 169 – 26x + y2 + 9 + 6y + z2 + 169 – 26z}


⇒ 9{x2 + 4x + y2 – 4y + z2 – 6z + 17} = 4{x2 – 26x + y2 + 6y + z2 – 26z + 347}


⇒ 9x2 + 36x + 9y2 – 36y + 9z2 – 54z + 153 = 4x2 – 104x + 4y2 + 24y + 4z2 – 104z + 1388


⇒ 9x2 + 36x + 9y2 – 36y + 9z2 – 54z + 153 – 4x2 + 104x – 4y2 – 24y – 4z2 + 104z – 1388 = 0


⇒ 5x2 + 5y2 + 5z2 + 140x – 60y + 50z – 1235 = 0


Hence locus of point P is 5x2 + 5y2 + 5z2 + 140x – 60y + 50z – 1235 = 0


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