If a = 5 + 2√6 and b = 1/a, then what will be the value of a²+b²?
Solution:
Given, a = 5 + 2√6
Also, b = 1/a
We have to find the value of a²+b².
Now, b = 1/(5+2√6)
To rationalise we have to take conjugate,
b = 1/(5+2√6) × (5-2√6)/(5-2√6)
By using algebraic identity,
(a - b)(a + b) = a² - b²
(5+2√6)(5-2√6) = (5)² - (2√6)²
= 25 - 4(6)
= 25 - 24
= 1
So, 1/(5+2√6) × (5-2√6)/(5-2√6) = (5-2√6)/(1)
b = (5-2√6)
a²+b² = (5+2√6)² + (5-2√6)²
By using algebraic identity,
(a + b)² = a² + 2ab + b²
(5+2√6)² = (5)² + 2(5)(2√6) + (2√6)²
= 25 + 20√6 + 4(6)
= 25 + 24 + 20√6
= 49 + 20√6
By using algebraic identity,
(a - b)² = a² + 2ab + b²
(5-2√6)² = (5)² - 2(5)(2√6) + (2√6)²
= 25 - 20√6 + 4(6)
= 25 + 24 - 20√6
= 49 - 20√6
Now, a²+b² = 49 + 20√6 + 49 - 20√6
= 49 + 49
= 98
Therefore, a²+b² = 98
✦ Try This: If a = 3 + 2√5 and b = 1/a, then what will be the value of a²+b²?
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 1
NCERT Exemplar Class 9 Maths Exercise 1.4 Sample Problem 1
If a = 5 + 2√6 and b = 1/a, then what will be the value of a²+b²
Summary:
If a = 5 + 2√6 and b = 1/a, then the value of a²+b² is 98
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