( a a + x )^ 1 2 + ( a a - x )^ 1 2 = - Vidyadip

MCQ

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

✓
$2 + \frac{{3{x^2}}}{{4{a^2}}} + ....$
B
$1 + \frac{{3{x^2}}}{{8{a^2}}} + ....$
C
$2 + \frac{x}{a} + \frac{{3{x^2}}}{{4{a^2}}} + ....$
D
$2 - \frac{x}{a} + \frac{{3{x^2}}}{{4{a^2}}}$+.....

✓

Answer

Correct option: A.

$2 + \frac{{3{x^2}}}{{4{a^2}}} + ....$

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

$ = [ {1 + ( { - \frac{1}{2}} )\,\left( {\frac{x}{a}} \right) + \frac{{\left( { - \frac{1}{2}} \right)\,\left( { - \frac{3}{2}} \right)}}{{2.1}}{{\left( {\frac{x}{a}} \right)}^2} + ....} ]$

$ + \left[ {1 + \left( { - \frac{1}{2}} \right)\,\left( { - \frac{x}{a}} \right) + \frac{{\left( { - \frac{1}{2}} \right)\,\left( { - \frac{3}{2}} \right)}}{{2.1}}{{\left( { - \frac{x}{a}} \right)}^2} + ....} \right]$

$ = 2 + \frac{{3{x^2}}}{{4{a^2}}} + $.........

Here odd terms cancel each other.

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