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Q.
Find the point on x-axis which is equidistant from the point $A(3,2, 2)$ and $B(5, 5,4)$.
Introduction to Three Dimensional Geometry
Solution:
The point on the $x$-axis is of the form $P(x, 0, 0)$. Since the points $A$ and $B$ are equidistant from $P$. Therefore $PA^2 = PB^2$
$\Rightarrow \left(x - 3\right)^{2} + \left(0- 2\right)^{2} + \left(0 - 2\right)^{2}$
$= \left(x - 5\right)^{2 }+ \left(0 - 5\right)^{2} + \left(0 - 4\right)^{2}$
$\Rightarrow 4x = 25 + 25 + 16 - 17$
i.e., $x = \frac{49}{4}$
Thus, the point $P$ on the $x$-axis $\left(\frac{49}{4}, 0, 0\right)$ is equidistant from $A$ and $B$.