If both a and b are rational numbers, find a and b respectively from the following.
$\frac{3 - \sqrt{5}}{3 + 2 \sqrt{5}} = a \sqrt{5} - b$

\frac{9}{11}, \frac{19}{11}

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\frac{19}{11}, \frac{9}{11}

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\frac{2}{11}, - \frac{19}{11}

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\frac{10}{11}, \frac{21}{11}

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Solution

The correct option is D $\frac{9}{11}, \frac{19}{11}$
$\frac{3 - \sqrt{5}}{3 + 2 \sqrt{5}} = \frac{3 - \sqrt{5}}{3 + 2 \sqrt{5}} \times \frac{3 - 2 \sqrt{5}}{3 - 2 \sqrt{5}}$
$= \frac{19 - 9 \sqrt{5}}{- 11}$
$= \frac{9 \sqrt{5}}{11} - \frac{19}{11}$
Compare with $a \sqrt{5} - b$ ,
$a = \frac{9}{11}, b = \frac{19}{11}$

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