Question

find the coordinates of a point which is equidistant from the points(a,0,0),(0,b,0),(0,0,c),(0,0,0)

Answer 1

AnswEr:

Let the coordinates of a point which is equidistant from the points be O (0,0,0) , A (a,0,0) , B (0,b,0) and C (0,0,c).

• Let P (x,y,z) be the required point. Then, OP = PA = PB = PC.

Now, OP = PA

$\\ \implies \sf {OP}^{2} = { PA}^{2} \\ \\ \\ \implies \sf {x}^{2} + {y}^{2} + {z}^{2} = {(x - a)}^{2} + {(y - 0)}^{2} \\ \sf \: + {(z - 0)}^{2} \\ \\ \\ \implies \sf \: 0 = - 2ax + {a}^{2} \\ \\ \\ \implies \sf \: x = \frac{a}{2}$

Similarly, OP = PB $\implies$ y = b/2 and OP = PC $\implies$ z = c/2.

Hence, the coordinates of the points are

$\: \\ \qquad \sf \: ( \frac{a}{2} , \frac{b}{2} , \frac{c}{2} )$