Find the coordinates of the point which is equidistant from the four points O(0, 0, 0), A(2, 0, 0), B(0, 3, 0) and C(0, 0, 8).
Given: Points are O(0, 0, 0), A(2, 0, 0), B(0, 3, 0) and C(0, 0, 8)
To find: the coordinates of point which is equidistant from the points
Let required point P(x, y, z)
According to question:
PA = PB = PC = PO
⇒ PA2 = PB2 = PC2 = PO2
Formula used:
Distance between any two points (a, b, c) and (m, n, o) is given by,
Therefore,
The distance between P(x, y, z) and O(0, 0, 0) is PO,
Distance between P(x, y, z) and A(2, 0, 0) is PA,
Distance between P(x, y, z) and B(0, 3, 0) is PB,
Distance between P(x, y, z) and C(0, 0, 8) is PC,
As PO2 = PA2
x2+ y2 + z2 = (x – 2)2 + y2 + z2
⇒ x2= x2+ 4 – 4x
⇒ 4x = 4
⇒ x = 1
As PO2 = PB2
x2+ y2 + z2 = x2+ (y – 3)2 + z2
⇒ y2= y2+ 9 – 6y
⇒ 6y = 9
As PO2 = PC2
x2+ y2 + z2 = x2 + y2 + (z – 8)2
⇒ z2= z2+ 64 – 16x
⇒ 16z = 64
⇒ z = 4
Hence point is equidistant from given points