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


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