Question

If \(\frac{{\sqrt 2 - 1}}{{\sqrt 2 + 1}} = a - b\sqrt 2 \), then what will be the value of a² + b²?
1. 9
2. 12
3. 10
4. 13

Answer 1

Correct Answer - Option 4 : 13

Given:

\(\frac{{√ 2 - 1}}{{√ 2 + 1}} = a - b√ 2 \)

Formula used:

(a – b)(a + b) = a² – b²

Calculations:

Multiply numerator and denominator by \({√ 2 - 1}\)

⇒ \(\frac{{√ 2 - 1}}{{√ 2 + 1}}\times \frac{{√ 2 - 1}}{{√ 2 - 1}} = a - b√ 2 \)

⇒ \(\frac{({√ 2 - 1})^2}{{(√ 2)^2 - 1^2}} = a - b√2\)

⇒ (2 + 1 – 2√2)/(2 – 1) =  a – b√2

⇒ 3 – 2√2 = a – b√2 

By comparing we get, a = 3, b = 2

a2 + b² = 3² + 2²

⇒ 9 + 4 = 13

∴ The value of a2 + b2  is 13