Correct Answer - Option 3 : 34
Given:
\(a = \dfrac{1}{3 - 2 \sqrt 2}\) and \(b= \dfrac{1}{3 + 2 \sqrt 2}\)
Concept:
Rational number: A number that can be expressed as the ratio of two integers called a rational number.
If p and q are integers then p/q is the rational number, where q can not be zero.
Rationalization: It is the process of eliminating irrational numbers of the denominator in the fraction.
for example, if \(\dfrac{1}{a + √ b}\) is given fraction, then we have to multiply by a - √b on both numerator and denominator.
Formula used:
(a2 - b2) = (a - b)(a + b)
(a + b)2 = a2 + b2 + 2ab
(a - b)2 = a2 + b2 - 2ab
Calculation:
We have given
\(a = \dfrac{1}{3 - 2 √ 2}\)
On rationalization of given fraction
\(\Rightarrow a = \left( \dfrac{1}{3 - 2 √ 2} \right)× \left(\dfrac{3 +2 √ 2}{3 + 2 √ 2}\right) \)
\(⇒ a = \dfrac{3 + 2 √ 2}{(3)^2 - (2 √ 2)^2} \)
We know that,
(a2 - b2) = (a - b)(a + b)
\(⇒ a = \dfrac{3 + 2√2}{9 - 8}\)
⇒ a = 3 + 2√2
⇒ a2 = (3 + 2√2)2
We know that,
(a + b)2 = a2 + b2 + 2ab
⇒ a2 = 32 + (2√2)2 + 2 × 3 × 2√2
⇒ a2 = 17 + 12√2 ---(1)
Similarly, we can find
b = 3 - 2√2, and
b2 = 17 - 12√2 ---(2)
Therefore,
a2 + b2 = 17 + 12√2 + 17 - 12√2
⇒ a2 + b2 = 34
Hence, the value of a2 + b2 is 34.
