A) \[{{(a+b+c)}^{2}}\]
B) \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\]
C) \[ab+bc+ca\]
D) \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}-ab-bc-ca\]
Correct Answer: A
Solution :
| [a] LHS\[=\frac{x-{{a}^{2}}}{b+c}+\frac{x-{{b}^{2}}}{b+c}+\frac{x-{{c}^{2}}}{a+b}\] From option putting \[x={{(a+b+c)}^{2}}\] \[\frac{{{(a+b+c)}^{2}}-{{a}^{2}}}{b+c}+\frac{{{(a+b+c)}^{2}}-{{b}^{2}}}{c+a}+\frac{{{(a+b+c)}^{2}}-{{c}^{2}}}{a+b}\] \[\frac{(a+b+c+a)(a+b+c-a)}{b+c}+\frac{(a+b+c+b)(a+b+c-b)}{c+a}\] \[+\frac{(a+b+c+c)(a+b+c-c)}{a+b}\] \[=(2a+b+c)+(a+2b+c)+(a+b+2c)\] \[=4\,(a+b+c)=\text{RHS}\] |
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