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The number of irrational solutions of $\sqrt{x^2+\sqrt{x^2+11}}+\sqrt{x^2-\sqrt{x^2+11}}=4$ are

$\begin{array}{1 1}(A)\;0&(B)\;1\\(C)\;2&(D)\;4\end{array} $

1 Answer

Put $\sqrt{x^2+11}=t$
Above equation can be written as,
$\sqrt{t^2+t-11}+\sqrt{t^2-t-11}=4$------(1)
$(t^2+t-11)+t^2-t-11=2t$--------(2)
Dividing (2) by (1) we get,
$\sqrt{t^2+t-11}-\sqrt{t^2-t-11}=\large\frac{t}{2}$------(3)
Adding (1) & (3) we get,
$2(\sqrt{(t^2+t-11})=4+\large\frac{t}{2}$
$\Rightarrow 4(t^2+t-11)=16+\large\frac{t^2}{4}$$+4t$
$\Rightarrow (t^2+t-11)=4+t+\large\frac{t^2}{16}$
$\Rightarrow t=4$
$\Rightarrow x=\pm \sqrt 5$
Hence (D) is the correct answer.
answered Apr 15, 2014 by sreemathi.v
 

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