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in 3D Coordinate Geometry by (55.0k points)
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Find the point on the z-axis which is equidistant from the points A(1, 5, 7) and B(5, 1, -4).

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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).

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