Determine the point on z-axis which is equidistant from the point… - Vidyadip

Question

Determine the point on $z-$axis which is equidistant from the points $(1, 5, 7)$ and $(5, 1, -4).$

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Answer

Let $P(0, 0, z)$ be the equidistant from $Q(1, 5, 7)$ and $R(5, 1, -4).$
So
$(PQ)^2 = (PR)^2 \Rightarrow (0 - 1)^2 + (0 - 5)^2 + (z - 7)^2 = (0 - 5) + (0 - 1)^2 + (z + 4)^2$
$\Rightarrow 1 + 25 + (z - 7)^2 = 25 + 1 + (z + 4)^2$
$\Rightarrow 26 + z^2 + 49 - 14z = 26 + z^2 + 8z + 16$
$\Rightarrow -14z - 8z = 16 - 49$
$\Rightarrow -22z = -33$
$\Rightarrow\text{z}=\frac{-33}{-22}$
$\Rightarrow\text{z}=\frac{3}{2}$
Required point $=\Big(0,\ 0,\ \frac{3}{2}\Big)$

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