QQuestion
If a = \frac{√5 + 1}{√5 - 1} and ab = 1 calculate (\frac{\text{a}}{\text{b}} + \frac{\text{b}}{\text{a}})
✓Answer
Given: a = \frac{√5 + 1}{√5 - 1}
⇒ a = \frac{√5 + 1}{√5 - 1} × \frac{√5 + 1}{√5 + 1}
⇒ a = \frac{(√5 + 1)²}{(√5)² - 1²}
⇒ a = \frac{(√5)² + 2√5 + (1)²}{(√5)² - 1²}
⇒ a = \frac{5 + 2√5 + 1}{5 - 1}
⇒ a = \frac{6 + 2√5}{4}
⇒ a = \frac{3 + √5}{2}
Now, ab = 1
⇒ b = \frac{1}{\text{a}}
⇒ b = \frac{2}{3 + √5}
⇒ b = \frac{2}{3 + √5} × \frac{3 - √5}{3 - √5}
⇒ b = \frac{2(3 - √5)}{(3)² - (√5)²}
⇒ b = \frac{2(3 - √5)}{4}
⇒ b = \frac{(3 - √5)}{2}
a + b = \frac{(3 + √5)}{2} + \frac{(3 - √5)}{2}
a + b = \frac{(3 + √5) + (3 - √5)}{2} = 3
a² + b² = (a + b)² – 2ab = 3² – 2 × 1 = 9
Now, (\frac{\text{a}}{\text{b}} + \frac{\text{b}}{\text{a}}) = (\frac{\text{a² + b²}}{\text{ab}})
= \frac{9}{1}
= 9
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