10th Class Mathematics Real Numbers Question Bank Real Numbers

  • question_answer The value of \[\frac{a+\sqrt{{{a}^{2}}-{{b}^{2}}}}{a-\sqrt{{{a}^{2}}-{{b}^{2}}}}+\frac{a-\sqrt{{{a}^{2}}-{{b}^{2}}}}{a+\sqrt{{{a}^{2}}-{{b}^{2}}}}\] is __.

    A) \[\frac{{{a}^{2}}}{{{b}^{2}}}\]                        

    B) \[\frac{{{b}^{2}}}{{{a}^{2}}}\]            

    C)  \[\frac{a}{b}\]                       

    D)         \[\frac{2(2{{a}^{2}}-{{b}^{2}})}{{{b}^{2}}}\]

    Correct Answer: D

    Solution :

    We have, \[\frac{a+\sqrt{{{a}^{2}}-{{b}^{2}}}}{a-\sqrt{{{a}^{2}}-{{b}^{2}}}}+\frac{a-\sqrt{{{a}^{2}}-{{b}^{2}}}}{a+\sqrt{{{a}^{2}}-{{b}^{2}}}}\] Rationalizing both terms, we get             \[\frac{{{\left( a+\sqrt{{{a}^{2}}-{{b}^{2}}} \right)}^{2}}}{{{a}^{2}}-{{a}^{2}}+{{b}^{2}}}+\frac{{{\left( a-\sqrt{{{a}^{2}}-{{b}^{2}}} \right)}^{2}}}{{{a}^{2}}-{{a}^{2}}+{{b}^{2}}}\]             \[=\frac{1}{{{b}^{2}}}\left[ \begin{align}   & {{a}^{2}}+{{a}^{2}}-{{b}^{2}}+2a\sqrt{{{a}^{2}}-{{b}^{2}}}+{{a}^{2}}+{{a}^{2}} \\  & -{{b}^{2}}-2a\sqrt{{{a}^{2}}-{{b}^{2}}} \\ \end{align} \right]\] \[=\frac{2}{{{b}^{2}}}(2{{a}^{2}}-{{b}^{2}})\]


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