Question

Find the value of $a$ in the following: $\frac{3-\sqrt{5}}{3+2\sqrt{5}}=a\sqrt{5}-\frac{19}{11}$

• A. $\frac{9}{11}$
• B. $\frac{2}{13}$
• C. $\frac{9}{42}$
• D. $\frac{11}{9}$

Answer 1

The correct option is A

$\frac{9}{11}$

Step1 : Rationalize the L.H.S. of the equation.

L.H.S. is Left Hand Side

$=\frac{\left(3-\sqrt{5}\right)\left(3-2\sqrt{5}\right)}{\left(3+2\sqrt{5}\right)\left(3-2\sqrt{5}\right)}$ [multiplying $3-2\sqrt{5}$with the numerator and denominator]

$=\frac{3\left(3-2\sqrt{5}\right)-\sqrt{5}\left(3-2\sqrt{5}\right)}{9-20}$ [$\left(a+b\right)\left(a-b\right)=a^2-b^2$ is used in denominator]

$$\begin{gathered} =\frac{9-6\sqrt{5}-3\sqrt{5}+10}{-11} \\ =\frac{19-9\sqrt{5}}{-11} \\ =\frac{9\sqrt{5}-19}{11} \\ =\frac{9\sqrt{5}}{11}-\frac{19}{11} \end{gathered}$$

Step2 : Compare L.H.S. with R.H.S. of the equation.

R.H.S. is Right Hand Side

L.H.S. = R.H.S.

$\frac{9\sqrt{5}}{11}-\frac{19}{11}=a\sqrt{5}-\frac{19}{11}$

From the above equation, we get $a\sqrt{5}=\frac{9\sqrt{5}}{11}$

Therefore, $a=\frac{9}{11}$

Hence, option A is the correct answer.