A) \[{{x}^{2}}+{{p}^{2}}x+{{P}^{2}}q=0\]
B) \[{{x}^{2}}-{{q}^{2}}x+{{P}^{2}}q=0\]
C) \[{{x}^{2}}+{{q}^{2}}x+{{P}^{2}}q=0\]
D) \[{{x}^{2}}-{{p}^{2}}x+{{P}^{2}}q=0\]
Correct Answer: D
Solution :
Since \[\alpha \] and \[\beta \] are roots of the equation\[{{x}^{2}}+px+q=0\], therefore \[\alpha +\beta =-p\] ? (i) and \[\alpha \beta =q\] ? (ii) Sum of the roots \[={{\alpha }^{2}}+\alpha \beta +{{\beta }^{2}}+\alpha \beta \] \[=(\alpha +\beta )={{p}^{2}}\] Product of the roots\[=(\alpha +\alpha \beta )({{\beta }^{2}}+\alpha \beta )\] \[=\alpha \beta {{(\alpha +\beta )}^{2}}=q{{p}^{2}}\] Required equation will be \[{{x}^{2}}-\](Sum of the roots) \[x+\] Product of the roots \[=\,\,0\] or \[{{x}^{2}}-{{p}^{2}}x+q{{p}^{2}}=0\]
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