If x = (√3+√2)/(√3-√2) and y = (√3-√2)/(√3+√2), then find the value of x²+y²
Solution:
Given, x = (√3+√2)/(√3-√2) and y = (√3-√2)/(√3+√2)
We have to find the value of x²+y².
x = (√3+√2)/(√3-√2)
Taking conjugate,
(√3+√2)/(√3-√2) = (√3+√2)/(√3-√2) × (√3+√2)/(√3+√2)
= (√3+√2)²/(√3-√2)(√3+√2)
By using algebraic identity,
(a - b)(a + b) = a² - b²
(√3-√2)(√3+√2) = (√3)² - (√2)²
= 3 - 2
= 1
So, (√3+√2)²/(√3-√2)(√3+√2) = (√3+√2)²/(1)
= (√3+√2)²
By using algebraic identity,
(a+b)² = a² + 2ab + b²
(√3+√2)² = (√3)² + 2(√3)(√2) + (√2)²
= 3 + 2√6 + 2
x = 5 + 2√6
x² = (5 + 2√6)²
= (5)² + 2(5)(2√6) + (2√6)²
= 25 + 20√6 + 4(6)
= 25 + 24 + 20√6
x² = 49 + 20√6
y = (√3-√2)/(√3+√2)
Taking conjugate,
(√3-√2)/(√3+√2) = (√3-√2)/(√3+√2) × (√3-√2)/(√3-√2)
= (√3-√2)²/(√3-√2)(√3+√2)
So, (√3-√2)²/(√3-√2)(√3+√2) = (√3-√2)²/(1)
y = (√3-√2)²
By using algebraic identity,
(a-b)² = a² - 2ab + b²
(√3-√2)² = (√3)² - 2(√3)(√2) + (√2)²
= 3 - 2√6 + 2
y = 5 - 2√6
y² = (5 - 2√6)²
= (5)² - 2(5)(2√6) + (2√6)²
= 25 - 20√6 + 4(6)
= 25 + 24 - 20√6
y² = 49 - 20√6
x²+y² = 49 + 20√6 + 49 - 20√6
= 49 + 49
= 98
Therefore, x²+y² = 98
✦ Try This: If x = (√5+√3)/(√5-√3) and y = (√5-√3)/(√5+√3), then the value of x²+y²
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 1
NCERT Exemplar Class 9 Maths Exercise 1.4 Problem 5
If x = (√3+√2)/(√3-√2) and y = (√3-√2)/(√3+√2), then find the value of x²+y²
Summary:
If x = (√3+√2)/(√3-√2) and y = (√3-√2)/(√3+√2), then the value of x²+y² is 98
☛ Related Questions:
visual curriculum