Given: Points are A(1, 5, 7), B(5, 1, -4)

To find: the point on z-axis which is equidistant from the points

As we know x = 0 and y = 0 on z-axis

Let R(0, 0, z) any point on z-axis

According to the question:

RA = RB

⇒ RA² = RB²

Formula used:

The distance between any two points (a, b, c) and (m, n, o) is given by,

Therefore,

Distance between R(0, 0, z) and A(1, 5, 7) is RA,

Distance between R(0, 0, z) and B(5, 1, -4) is RB,

As RA² = RB²

26+ (z – 7)² = (z + 4)² + 26

⇒ z²+ 49 – 14z + 26 = z²+ 16 + 8z + 26

⇒ 49 – 14z = 16 + 8z

⇒ 49 – 16 = 14z + 8z

⇒ 22z = 33

Hence point on z-axis is equidistant from (1, 5, 7) and (5, 1, -4)