First slide

Introduction to 3-D Geometry

Question

The equation of the set of points which are equidistant from the points (1,2,3) and (3,2, - 1) is

Easy

A

x -3z =0
B

x -2z =0
C

x-4z=0
D

x+2z=0

Solution

Let the given points are A and B.
Let P(x, y, z) be any point equidistant from A and B.
PA = PB,
i.e. Distance between P and A = distance between P and B
$\begin{array}{l} \Rightarrow \sqrt{(x - 1)^{2} + (y - 2)^{2} + (z - 3)^{2}} = \sqrt{(x - 3)^{2} + (y - 2)^{2} + (z + 1)^{2}} . \\ \Rightarrow (x - 1)^{2} + (y - 2)^{2} + (z - 3)^{2} = (x - 3)^{2} + (y - 2)^{2} + (z + 1)^{2} \\ \Rightarrow x^{2} + 1 - 2 x + y^{2} + 4 - 4 y + z^{2} + 9 - 6 z \\ = x^{2} + 9 - 6 x + y^{2} + 4 - 4 y + z^{2} + 1 + 2 z \\ \Rightarrow 4 x - 8 z = 0 \\ \Rightarrow x - 2 z = 0 \end{array}$
which is the required equation.

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