A) \[\left( \frac{a}{4},\frac{b}{4},\frac{c}{4} \right)\]
B) \[\left( \frac{a}{2},\frac{b}{4},\frac{c}{4} \right)\]
C) \[\left( \frac{a}{2},\frac{b}{2},\frac{c}{2} \right)\]
D) (a, b, c)
Correct Answer: C
Solution :
The required point is the centre of the sphere through the given points. Let the equation of sphere be \[{{x}^{2}}+{{y}^{2}}+2ux+2vy+2wz+d=0\] .....(i) Sphere (i) is passing through (0, 0, 0), (a, 0, 0), (0, b, 0) and (0, 0, c), \[\therefore \,\,d=0\] \[{{a}^{2}}+2ua=0\Rightarrow u=-a/2\] \[{{b}^{2}}+2vb=0\,\,\Rightarrow \,\,v=-b/2\] \[{{c}^{2}}+2wc=0\,\Rightarrow w=-c/2\] Therefore, centre of sphere is \[(a/2,\,b/2,\,c/2)\], which is also the required point.
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