A) \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-x+y-z=1\]
B) \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-6x+2y-4z=1\]
C) \[2{{x}^{2}}+2{{y}^{2}}+2{{z}^{2}}-6x+2y-4z-15=0\]
D) \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-3x+y-2z=1\]
E) \[2{{x}^{2}}+2{{y}^{2}}+2{{z}^{2}}-6x+2y-4z-25=0\]
Correct Answer: E
Solution :
Equation of sphere is \[2{{x}^{2}}+2{{y}^{2}}+2{{z}^{2}}-6x-2y-4z-1=0\] Radius of sphere is \[\sqrt{\frac{9}{4}+\frac{1}{4}+\frac{4}{4}+\frac{1}{2}}=2\] Equation of family of concentric sphere is \[{{x}^{2}}+{{y}^{2}}+\text{ }{{z}^{2}}-3x+\text{ }y-2z+\lambda =0\] ...(i) \[\therefore \]According to question, \[\sqrt{\frac{9}{4}+\frac{1}{4}+1-\lambda }=4\] \[\Rightarrow \] \[\frac{14}{4}-\lambda =16\] \[\Rightarrow \] \[\lambda =\frac{14}{4}-16=-\frac{25}{2}\] \[\therefore \]From Eq. (i) \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-3x+y-2z-\frac{25}{2}=0\] \[\Rightarrow \]\[2{{x}^{2}}+2{{y}^{2}}+2{{z}^{2}}-6x+2y-4z-25=0\]
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