If $$\frac{\sqrt{5}-1}{\sqrt{5}+1} = a + b\sqrt{5}$$​, then find the value of a and b.

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Correct option is A

Given:

$$\frac{\sqrt{5}-1}{\sqrt{5}+1} = a + b\sqrt{5},$$

Solution:

$$\frac{\sqrt{5}-1}{\sqrt{5}+1} = a + b\sqrt{5},$$

Taking L.H.S.

$$\frac{\sqrt{5}-1}{\sqrt{5}+1}$$$$\times \frac{\sqrt{5}-1}{\sqrt{5}-1}$$

= $$\frac{(\sqrt{5}-1)^2 }{5-1}$$

= $$\frac {5+ 1- 2\sqrt5}{4}$$

=$$\frac {6}{4} - \frac{2\sqrt5}{4}$$

=$$\frac{3}{2} - \frac{\sqrt5}{2}$$​

Then,

$$\frac{3}{2} - \frac{\sqrt5}{2}$$ = $$a + b\sqrt{5}$$

Comparing both sides,

​$$a = \frac{3}{2}, b = \left(-\frac{1}{2}\right)$$​​

Option (a) is right answer.