Question

find the value of a if 3-root5/3+2root5=aroot5-19/11

Answer 1

Hi...☺

Here is your answer...✌

GIVEN THAT,

$\frac{3 - \sqrt{5} }{ 3 + 2 \sqrt{5} } = \frac{a \sqrt{5} - 19}{11} \: \: \: \: \: \: \: \: \: \: .....(1)$
On rationalising LHS
We get,

LHS

$= \frac{3 - \sqrt{5} }{3 + 2 \sqrt{5} } \times \frac{3 - 2 \sqrt{5} }{3 - 2 \sqrt{5} } \\ \\ = \frac{9 - 6 \sqrt{5} - 3 \sqrt{5} + 10}{ {3}^{2} - {(2 \sqrt{5} )}^{2} } \\ \\ = \frac{19 - 9 \sqrt{5} }{9 - 20} \\ \\ = \frac{19 - 9 \sqrt{5} }{ - 11} \\ \\ = \frac{9 \sqrt{5} - 19}{11}$

Now

From (1) we have,

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