A) \[-6\]
B) \[-4\]
C) \[0\]
D) \[4\]
Correct Answer: B
Solution :
[b] \[\alpha \]and \[\beta \] are root of \[{{x}^{2}}-bx=b.\] \[\therefore \,\,\,\,\alpha +\beta =b,\,\,\alpha \beta =-b\] \[|\alpha |\] and \[|\beta |\] are roots of \[{{x}^{2}}+px+q=0.\] \[\therefore \,\,|\alpha |+|\beta |=-p,\] \[\,|\alpha ||\beta |=q\] \[\therefore \,\,\,\,{{p}^{2}}-8q=|\alpha {{|}^{2}}+|\beta {{|}^{2}}-6|\alpha ||\beta |\] \[={{(\alpha +\beta )}^{2}}-2\alpha \beta -6|\alpha ||\beta |\] \[={{b}^{2}}+2b++6(-b)=({{b}^{2}}-4b)={{(b-2)}^{2}}-4\] So, minimum value of \[({{p}^{2}}-8q)\] is\[-4\].
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