Question

If both '$a$, and '$b$' are rational numbers and $\frac{3-\sqrt{5}}{3+2\sqrt{5}}=a\sqrt{5}-b$, then the value of a and b respectively are ___.

• A. $\frac{9}{11},\frac{19}{11}$
• B. $\frac{19}{11},\frac{10}{11}$
• C. $\frac{2}{11},\frac{8}{11}$
• D. $\frac{10}{11},\frac{21}{11}$

Answer 1

The correct option is A $\frac{9}{11},\frac{19}{11}$

On multiplying and dividing with $3-2\sqrt{5}$, We get

$\frac{3-\sqrt{5}}{3+2\sqrt{5}}\times \frac{3-2\sqrt{2}}{3-2\sqrt{5}}$

$=\frac{(3-\sqrt{5})(3-2\sqrt{5})}{9-20}=\frac{9+10-9\sqrt{5}}{-11}$

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

$a=\frac{9}{11}$ and $b=\frac{19}{11}$