A) \[{{K}^{2}}-109\]
B) \[2{{K}^{2}}-109\]
C) \[3{{K}^{2}}-109\]
D) \[4{{K}^{2}}-10\]
Correct Answer: B
Solution :
[b] Let the coordinates of point P be (x, y, z). Here, \[P{{A}^{2}}={{(x-3)}^{2}}+{{(y-4)}^{2}}+{{(z-5)}^{2}}\] \[P{{B}^{2}}={{(x+1)}^{2}}+{{(y-3)}^{2}}+{{(z+7)}^{2}}\] By the given condition \[P{{A}^{2}}+P{{B}^{2}}=2{{K}^{2}}\] We have \[{{(x-3)}^{2}}+{{(y-4)}^{2}}+{{(z-5)}^{2}}+{{(x+1)}^{2}}\] \[+{{(y-3)}^{2}}+{{(z+7)}^{2}}=2{{K}^{2}}\] i.e. \[2{{x}^{2}}+2{{y}^{2}}+2{{z}^{2}}-4x-14y+4z=2{{K}^{2}}-109\]
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