Find the values of a and b in (7+√5)/(7-√5) - (7-√5)/(7+√5) = a + 7√5b/11
Solution:
Given, the expression is (7+√5)/(7-√5) - (7-√5)/(7+√5) = a + 7√5b/11
We have to find the values of a and b.
Considering LHS,
LHS: (7+√5)/(7-√5) - (7-√5)/(7+√5)
[(7+√5)/(7+√5) - (7-√5)/(7-√5)] / (7+√5)(7-√5) = [(7+√5)² - (7-√5)²] / (7+√5)(7-√5)
By using algebraic identity,
(a² - b²) = (a - b)(a + b)
(7+√5)(7-√5) = (7)² - (√5)²
= 49 - 5
= 44
So, [(7+√5)² - (7-√5)²] / (7+√5)(7-√5) = [(7+√5)² - (7-√5)²] / 44
By using algebraic identity,
(a + b)² = a² + 2ab + b²
(7 + √5)² = (7)² + 2(7)(√5) + (√5)²
= 49 + 14√5 + 5
= 54 + 14√ 5
By using algebraic identity,
(a - b)² = a² - 2ab + b²
(7 - √5)² = (7)² - 2(7)(√5) + (√5)²
= 49 - 14√5 + 5
= 54 - 14√ 5
So, [(7+√5)² - (7-√5)²] = (54 + 14√ 5) - (54 - 14√ 5)
= 54 + 14√ 5 - 54 + 14√ 5
= 28√5
Now, [(7+√5)² - (7-√5)²] / 44 = 28√5/44
= 14√5/22
= 7√5/11
So, a + 7√5b/11 = 7√5/11
a = 0
b = 1
Therefore, the values of a and b are 0 and 1.
✦ Try This: Find the values of a² - b² if a = 1/(7 - 4√3) and b = 1/(7 + 4√3)
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 1
NCERT Exemplar Class 9 Maths Exercise 1.3 Problem 11(iv)
Find the values of a and b in (7+√5)/(7-√5) - (7-√5)/(7+√5) = a + 7√5b/11
Summary:
The values of a and b in (7+√5)/(7-√5) - (7-√5)/(7+√5) = a + 7√5b/11 are 0 and 1 respectively
☛ Related Questions:
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