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Question: If x is very small in magnitude compared with a such that \(\left( \frac{a}{a + x} \right)^{1/2}\)+\...

If x is very small in magnitude compared with a such that (aa+x)1/2\left( \frac{a}{a + x} \right)^{1/2}+(aa−x)1/2\left( \frac{a}{a - x} \right)^{1/2}= 2 + k x2a2\frac{x^{2}}{a^{2}}, then the value of k is –

A

14\frac{1}{4}

B

12\frac{1}{2}

C

34\frac{3}{4}

D

1

Answer

34\frac{3}{4}

Explanation

Solution

(aa+x)1/2\left( \frac{a}{a + x} \right)^{1/2}+(aa−x)1/2\left( \frac{a}{a - x} \right)^{1/2}

=1(1+xa)1/2\frac{1}{\left( 1 + \frac{x}{a} \right)^{1/2}}+1(1−xa)1/2\frac{1}{\left( 1 - \frac{x}{a} \right)^{1/2}}

=(1+xa)–1/2\left( 1 + \frac{x}{a} \right)^{–1/2}+ (1+(−xa))–1/2\left( 1 + \left( - \frac{x}{a} \right) \right)^{–1/2}

= (1+(−12)(xa)+(−12)(−32)1.2(xa)2)\left( 1 + \left( - \frac{1}{2} \right)\left( \frac{x}{a} \right) + \frac{\left( - \frac{1}{2} \right)\left( - \frac{3}{2} \right)}{1.2}\left( \frac{x}{a} \right)^{2} \right)

+ (1+(−12)(−xa)+(−12)(−32)1.2(−xa)2)\left( 1 + \left( - \frac{1}{2} \right)\left( - \frac{x}{a} \right) + \frac{\left( - \frac{1}{2} \right)\left( - \frac{3}{2} \right)}{1.2}\left( - \frac{x}{a} \right)^{2} \right)

(Neglecting(xa)3\left( \frac{x}{a} \right)^{3}and higher powers)

= (1−x2a+3x28a2)\left( 1 - \frac{x}{2a} + \frac{3x^{2}}{8a^{2}} \right)+(1+x2a+3x28a2)\left( 1 + \frac{x}{2a} + \frac{3x^{2}}{8a^{2}} \right)= 2 + 3x24a2\frac{3x^{2}}{4a^{2}}

\ k = 34\frac{3}{4}.