Question

If 3+ root5
\2 root5+3

= + 5 , find the value of rational numbers a and b.​

Answer 1

Answer:

Answer:

a = 9 and b = 19

Step-by-step explanation:

Given that :

\frac{3-\sqrt 5}{3+2\sqrt 5}=\frac{a\sqrt 5-b}{11}

3+2

5

3−

5

=

11

a

5

−b

Now, Taking L.H.S. first and the rationalizing the denominator we get :

\begin{gathered}\implies \frac{3-\sqrt 5}{3+2\sqrt 5}\\\\\implies \frac{3-\sqrt 5}{3+2\sqrt 5}\times \frac{3-2\sqrt 5}{3-2\sqrt 5}\\\\\implies \frac{9-6\sqrt 5-3\sqrt 5+2\times 5}{9-20}\\\\ \implies \frac{19-9\sqrt{5}}{-11}\\\\\implies \frac{9\sqrt 5-19}{11}\\\\\implies \frac{9\sqrt 5}{11}-\frac{19}{11}\end{gathered}

⟹

3+2

5

3−

5

⟹

3+2

5

3−

5

×

3−2

5

3−2

5

⟹

9−20

9−6

5

−3

5

+2×5

⟹

−11

19−9

5

⟹

11

9

5

−19

⟹

11

9

5

−

11

19

Now, Comparing the above expression with the R.H.S. We get ,

a = 9 and b = 19