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
$(P Q)^2=(P R)^2 \Rightarrow(0-1)^2+(0-5)^2+(z-7)^2=(0-5)+(0-1)^2+(z+4)^2$
$\Rightarrow 1+25+(z-7)^2=25+1+(z+4)^2$
$\Rightarrow 26+z^2+49-14 z=26+z^2+8 z+16$
$\Rightarrow-14 z-8 z=16-49$
$\Rightarrow-22 z=-33$
$\Rightarrow z=\frac{-33}{-22}$
$\Rightarrow z=\frac{3}{2}$
$\text { Required point }=\left(0,0, \frac{3}{2}\right)$

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