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Question
Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
Solution
Let a point P(x, y, z) be equidistant from point A(1, 2, 3) and point B(3, 2, – 1).
AB = `sqrt((x - 1)^2 + (y - 2)^2 + (z - 3)^2)`
AC = `sqrt((x - 3)^2+ (y - 2)^2 + (z + 1)^2)`
It is given that AB= AC
= `sqrt((x - 1)^2 + (y - 2)^2 + (z - 3)^2)` = `sqrt((x - 3)^2+ (y - 2)^2 + (z + 1)^2)`
= (x – 1)2 + (y – 2)2 + (z – 3)2 = (x – 3)2+ (y – 2)2 + (z + 1)2
= (x2 – 2x + 1) + (z2 – 6z + 9) = (x2 – 6x + 9) + (z2 + 2z + 1)
= -2x - 6z + 10 = -6x + 2z + 10
= - 2x - 6z + 6x - 2z = 0
= 4x - 8z = 0
Hence, Required equation = x – 2z = 0
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