A) \[\left( \frac{a}{2},0,0 \right)\]
B) \[\left( 0,\frac{a}{2},0 \right)\]
C) \[\left( 0,0,\frac{a}{2} \right)\]
D) \[\left( \frac{a}{2},\,\frac{b}{2},\,0 \right)\]
E) \[\left( \frac{a}{2},\frac{a}{2},\frac{c}{2} \right)\]
Correct Answer: E
Solution :
Let the equation of sphere be \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2ux+2vy+2wz+d=0\] It passes through origin \[\therefore \] \[d=0\] Also it passes through (a, 0, 0), (0, b, 0), (0, 0, c) \[\Rightarrow \] \[{{a}^{2}}+2ua=0\] \[\Rightarrow \] \[{{a}^{2}}=-2ua\Rightarrow u=-\frac{a}{2}\] Similarly, \[v=-\frac{b}{2},w=-\frac{c}{2}\] \[\therefore \]Centre \[(-\text{ }u,-\text{ }v,-w)=\left( \frac{a}{2},\frac{b}{2},\frac{c}{2} \right)\].
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