A) \[1+\frac{1}{2}\frac{x}{a}\]
B) \[\frac{x}{a}\]
C) \[1+\frac{3}{4}\frac{{{x}^{2}}}{{{a}^{2}}}\]
D) \[2+\frac{3}{4}\frac{{{x}^{2}}}{{{a}^{2}}}\]
Correct Answer: D
Solution :
[d] \[{{\left( \frac{a}{a+x} \right)}^{\frac{1}{2}}}+{{\left( \frac{a}{a-x} \right)}^{\frac{1}{2}}}\] \[={{\left( \frac{a+x}{a} \right)}^{-\frac{1}{2}}}+{{\left( \frac{a-x}{a} \right)}^{-\frac{1}{2}}}\] \[={{\left( 1+\frac{x}{a} \right)}^{-\frac{1}{2}}}+{{\left( 1-\frac{x}{a} \right)}^{-\frac{1}{2}}}\] \[=\left[ 1-\frac{1}{2}\frac{x}{a}+\frac{3}{8}\frac{{{x}^{2}}}{{{a}^{2}}} \right]+\left[ 1+\frac{1}{2}\frac{x}{a}+\frac{3}{8}\frac{{{x}^{2}}}{{{a}^{2}}} \right]\] \[\left[ \because \,\,\,\,\,x<<a,\,\,\therefore \frac{x}{a}<<1 \right]=2+\frac{3}{4}.\frac{{{x}^{2}}}{{{a}^{2}}}\]
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