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Question
Find the equation of the set of the points P such that its distances from the points A(3, 4, –5) and B(–2, 1, 4) are equal.
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Solution
Let P (x, y, z) be any point which is equidistant from A (3,4,5) and B (\[-\]2,1,4) .Then,
PA = PB\[\Rightarrow \sqrt{\left( x - 3 \right)^2 + \left( y - 4 \right)^2 + \left( z + 5 \right)^2} = \sqrt{\left( x + 2 \right)^2 + \left( y - 1 \right)^2 + \left( z - 4 \right)^2}\]
\[ \Rightarrow \sqrt{x^2 - 6x + 9 + y^2 - 8y + 16 + z^2 + 10z + 25} = \sqrt{x^2 + 4x + 4 + y^2 - 2y + 1 + z^2 - 8z + 16}\]
\[ \Rightarrow x^2 - 6x + 9 + y^2 - 8y + 16 + z^2 + 10z + 25 = x^2 + 4x + 4 + y^2 - 2y + 1 + z^2 - 8z + 16\]
\[ \Rightarrow - 10x - 6y + 18z + 29 = 0\]
\[ \therefore 10x + 6y - 18z - 29 = 0\]
Hence, 10x + 6y\[-\]18z\[-\]29 = 0 is the required equation.
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