The locus of a point P which moves such that P A^{2} - P B^{2} = 2 k^{2} where A and B are \left(\right. 3 , 4 , 5 \left.\right) and \left(\right. - 1 , 3 ,

Engineering

Mathematics

Line of Greatest Slope

Question

The locus of a point P which moves such that

P A^{2} - P B^{2} = 2 k^{2}

where A and B are

(3, 4, 5)

and

(- 1, 3, - 7)

respectively is

8 x + 2 y + 24 z - 9 + 2 k^{2} = 0

8 x + 2 y + 24 z - 2 k^{2} = 0

8 x + 2 y + 24 z + 9 + 2 k^{2} = 0

8 x - 2 y + 24 z - 2 k^{2} = 0

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P A^{2} - P B^{2} = 2 k^{2}

\Rightarrow [{(x - 3)}^{2} + {(y - 4)}^{2} + {(z - 5)}^{2}] - [{(x + 1)}^{2} + {(y - 3)}^{2} + {(z + 7)}^{2}] = 2 k^{2}

\Rightarrow (x - 3 + x + 1) (x - 3 - x - 1) + (y - 4 + y - 3) (y - 4 - y + 3) + (z - 5 + z + 7) (z - 5 - z - 7) = 2 k^{2}

\Rightarrow (2 x - 2) (- 4) + (2 y - 7) (- 1) + (2 z + 1) (- 12) = 2 k^{2}

\Rightarrow - 8 x + 8 - 2 y + 7 - 24 z - 24 = 2 k^{2}

\Rightarrow - 8 x - 2 y - 24 z - 9 = 2 k^{2}

\Rightarrow 8 x + 2 y + 24 z + 9 + 2 k^{2} = 0

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