Consider, C(0,0,z) point which lies on z axis and is equidistant from points A(1, 5, 7) and B(5, 1, -4).
∴ AC = BC
\(\sqrt{(0-1)^2+(0-5)^2+(z-7)^2}\) = \(\sqrt{(0-5)^2+(0-1)^2+(z+4)^2}\)
Squaring both sides,
(0 - 1)2 + (0 - 5)2 + (z - 7)2 = (0 - 5)2 + (0 - 1)2 + (z + 4)2
1 + 25 + z2 -14z + 49 = 25 + 1 + z2 + 8z +16
-22z = -33 Z
= 1.5
The point C is (0,0,1.5).