A) \[\frac{{{p}^{2}}+{{q}^{2}}}{2}\]
B) -\[\frac{({{p}^{2}}+{{q}^{2}})}{2}\]
C) \[\frac{{{p}^{2}}-{{q}^{2}}}{2}\]
D) -\[\frac{({{p}^{2}}-{{q}^{2}})}{2}\]
Correct Answer: B
Solution :
Given equation can be written as \[{{x}^{2}}+x(p+q-2r)+pq-pr-qr=0\] .....(i) whose roots are \[\alpha \]and \[-\alpha \], then the product of roots \[-{{\alpha }^{2}}=pq-pr-qr=pq-r(p+q)\] .....(ii) and sum \[0=p+q-2r\,\,\,\Rightarrow r=\frac{p+q}{2}\] .....(iii) From (ii) and (iii), we get \[-{{\alpha }^{2}}=pq-\frac{p+q}{2}(p+q)=-\frac{1}{2}\left\{ {{(p+q)}^{2}}-2pq \right\}\] \[=-\frac{({{P}^{2}}+{{q}^{2}})}{2}\].
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