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જો $a $ અને $b$ વચ્ચેના સમગુણોત્તર મધ્યક $H$ હોય, તો $\frac{1}{{H\, - \,a}}\, + \,\frac{1}{{H - b}}$ નું મૂલ્ય કેટલું થાય ?
$a + b$
$ab$
$\frac{1}{a} + \frac{1}{b}$
$\frac{1}{a} - \frac{1}{b}$
Solution
$H\,\, = \,\,\frac{{2ab}}{{a\,\, + \,\,b}}$ મુક્તા $\frac{{\text{1}}}{{{\text{H}}\,\,{\text{ – }}\,\,{\text{a}}}}\,\, + \,\,\frac{1}{{H\,\, – \,\,b}}$ મળે.
$ = \,\,\frac{{\text{1}}}{{\left( {\frac{{{\text{2ab}}}}{{a\, + \,b}}\,\, – \,\,a} \right)}}\,\, + \,\,\frac{1}{{\left( {\frac{{2ab}}{{a\, + \,b}}\, – \,\,b} \right)}}\,\,\, = \,\,\frac{{a\,\, + \,\,b}}{{ab\,\, – \,\,{a^2}}}\,\, + \,\,\frac{{a\,\, + \,\,b}}{{ab\,\, – \,\,{b^2}}}$
$ = \,\,\left( {\frac{{a\,\, + \,\,b}}{{b\,\, – \,\,a}}} \right)\,\,\,\left( {\frac{{b\,\, – \,\,a}}{{ab}}} \right)\,\,\, = \,\,\frac{{a\,\, + \,\,b}}{{ab}}\,\,\, = \,\,\frac{1}{a}\,\, + \,\,\frac{1}{b}$