Determine the point on z-axis which is equidistant from the points (1, 5, 7) and (5, 1, -4)
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)