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, \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-3x+y-2z-\frac{1}{2}=0\] \ Centre of sphere \[=\left( \frac{3}{2},\frac{-1}{2},1 \right)\] Radius \[=\sqrt{\left( \frac{9}{4} \right)+\left( \frac{1}{4} \right)+1+\frac{1}{2}}=2\] Now, radius of required sphere = 4, which is concentric with the given sphere. Hence, equation of required sphere is, \[{{\left( x-\frac{3}{2} \right)}^{2}}+{{\left( y+\frac{1}{2} \right)}^{2}}+{{(z-1)}^{2}}=16\] i.e., \[2{{x}^{2}}+2{{y}^{2}}+2{{z}^{2}}-6x+2y-4z-25=0\].
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