A) \[2\{{{p}^{2}}-2q+{{{p}'}^{2}}-2{q}'-p{p}'\}\]
B) \[2\{{{p}^{2}}-2q+{{{p}'}^{2}}-2{q}'-q{q}'\}\]
C) \[2\{{{p}^{2}}-2q-{{{p}'}^{2}}-2{q}'-p{p}'\}\]
D) \[2\{{{p}^{2}}-2q-{{{p}'}^{2}}-2{q}'-q{q}'\}\]
Correct Answer: A
Solution :
As given, \[\alpha +\beta =p,\]\[\alpha \beta =q,{\alpha }'+{\beta }'={p}',{\alpha }'{\beta }'=q'\] Now, \[{{(\alpha -\alpha ')}^{2}}+{{(\beta -\alpha ')}^{2}}+{{(\alpha -{\beta }')}^{2}}+{{(\beta -{\beta }')}^{2}}\] \[=2({{\alpha }^{2}}+{{\beta }^{2}})+2(\alpha {{'}^{2}}+\beta {{'}^{2}})-2\alpha '(\alpha +\beta )-2\beta '(a+\beta )\] \[=2\left\{ {{(\alpha +\beta )}^{2}}-2\alpha \beta +{{({\alpha }'+{\beta }')}^{2}}-2\alpha '\beta '-(\alpha +\beta )({a}'+\beta ') \right\}\] \[=2\left\{ {{p}^{2}}-2q+{{{{p}'}}^{2}}-2{q}'-p{p}' \right\}\].
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