A) \[\frac{{{a}^{2}}-{{c}^{2}}}{4}\] and \[\frac{{{a}^{2}}}{4}\]
B) \[\frac{{{a}^{2}}+{{c}^{2}}}{4}\] and \[\frac{{{a}^{2}}}{4}\]
C) \[\frac{{{a}^{2}}-{{c}^{2}}}{2}\] and \[\frac{{{a}^{2}}}{4}\]
D) None of these
Correct Answer: A
Solution :
Given roots are real and distinct, then \[{{\operatorname{a}}^{2}}-4b>0\] \[\Rightarrow \,\,\,\operatorname{b}<{{a}^{2}}/4\] Again \[\alpha \] and \[\beta \] differ by a quantity less than \[c\left( c >0 \right)\] \[\Rightarrow \,\,\,\left| \alpha -\beta \right|<c\,\,or\,\,{{(\alpha -\beta )}^{2}}<{{c}^{2}}\] \[\Rightarrow \,\,\,{{\left( \alpha +\beta \right)}^{2}}-4\alpha \beta <{{c}^{2}} or {{a}^{2}}-4b<{{c}^{2}}\] or \[\frac{{{a}^{2}}-{{c}^{2}}}{4}<b\] \[\Rightarrow \,\,\,\frac{{{a}^{2}}-{{c}^{2}}}{4}<b<\frac{{{a}^{2}}}{4}\] by (1) and (2)
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