A) \[(4,\text{ }3,\text{ }5)\]
B) \[(4,9,-3)\]
C) \[(4,\text{ }3,-3)\]
D) \[(4,-9,3)\]
Correct Answer: B
Solution :
Given, equation of sphere is \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-6x-12y-2z+20=0\] ...(i) (2, 3, 5) is one end of the diameter. Let coordinates of second end are\[({{x}_{1}},{{y}_{1}},{{z}_{1}})\]and equation of sphere, \[(x-2)(x-{{x}_{1}})+(y-3)(y-{{y}_{1}})\] \[+(x-5)(z-{{z}_{1}})=0\] \[\Rightarrow \] \[{{x}^{2}}-2x-x{{x}_{1}}+2{{x}_{1}}+{{y}^{2}}-3y-y{{y}_{1}}+3{{y}_{1}}\] \[+{{z}^{2}}-5z-z{{z}_{1}}+5{{z}_{1}}=0\] \[\Rightarrow \] \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-({{x}_{1}}+2)x-({{y}_{1}}+3)y\] \[-({{z}_{1}}+5)z+2{{x}_{}}+3{{y}_{1}}+5{{z}_{1}}=0\] On comparing with Eq. (i), we get \[{{x}_{1}}+2=6\] \[\Rightarrow \] \[{{x}_{1}}=6-2=4\] \[{{y}_{1}}+3=12\] \[\Rightarrow \] \[{{y}_{1}}=9\] \[{{z}_{1}}+5=2\Rightarrow {{z}_{1}}=-3\] and \[2{{x}_{1}}+3{{y}_{1}}+5{{z}_{1}}=20\] Hence, coordinates of other end is\[(4,9,-3)\].
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