If \(a^2-16=-\displaystyle\frac{2}{a^2}\), then find the value of \(\sqrt{\displaystyle\frac{156a^2\times250}{13a^4+104a^2+26}}\).
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(5\sqrt{5}\)
The condition gives the value of \(a^2+\frac{2}{a^2}\). Dividing the numerator and the denominator of the fraction by \(13a^2\) brings out the same expression \(a^2+\frac{2}{a^2}\) in the denominator.
From the condition: \(a^2+\displaystyle\frac{2}{a^2}=16\)
Denominator: \(13a^4+104a^2+26=13a^2\left(a^2+8+\displaystyle\frac{2}{a^2}\right)\)
Numerator: \(156a^2\times250=13a^2\times12\times250=13a^2\times3000\)
Fraction \(\displaystyle=\frac{3000}{a^2+\frac{2}{a^2}+8}=\frac{3000}{16+8}=\frac{3000}{24}=125\)
\(\sqrt{125}=\sqrt{25\times5}=5\sqrt{5}\)