A) \[\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{2{{\Delta }^{2}}}\]
B) \[\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{{{\Delta }^{2}}}\]
C) \[\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{3{{\Delta }^{2}}}\]
D) None of these
Correct Answer: B
Solution :
\[\frac{1}{r_{1}^{2}}+\frac{1}{r_{2}^{2}}+\frac{1}{r_{3}^{2}}+\frac{1}{{{r}^{2}}}\] \[=\frac{{{(s-a)}^{2}}}{{{\Delta }^{2}}}+\frac{{{(s-b)}^{2}}}{{{\Delta }^{2}}}+\frac{{{(s-c)}^{2}}}{{{\Delta }^{2}}}+\frac{{{s}^{2}}}{{{\Delta }^{2}}}\] \[=\frac{1}{{{\Delta }^{2}}}[4{{s}^{2}}-2s(a+b+c)+{{a}^{2}}+{{b}^{2}}+{{c}^{2}}]\] \[=\frac{1}{{{\Delta }^{2}}}[4{{s}^{2}}-2s.2s+{{a}^{2}}+{{b}^{2}}+{{c}^{2}}]\] \[=\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{{{\Delta }^{2}}}\]
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