Question

If x = (√3 + √2)/( √3 – √2) and y = (√3 – √2)/( √3 + √2), then find the value of x² + y²?
1. 102
2. 90
3. 98
4. 96

Answer 1

Correct Answer - Option 3 : 98

Given:

x = (√3 + √2)/( √3 – √2)

 y = (√3 – √2)/( √3 + √2)

Concept:

First rationalize the values of ‘x’ and ‘y’ and then proceed.

Formula used:

(a + b)² = a² + b² + 2ab

(a + b)(a – b) = a² – b²

Calculation:

∵ x = (√3 + √2)/( √3 – √2)

⇒ x = [(√3 + √2)( √3 + √2)]/[( √3 + √2)( √3 – √2)]

⇒ x = [(√3 + √2)²]/[(√3)² – (√2)²]

⇒ x = [3 + 2 + 2√6]/(3 - 2)

⇒ x = (5 + 2√6)

Similarly;

y = (√3 - √2)/( √3 + √2)

⇒ y = [(√3 - √2)( √3 - √2)]/[( √3 + √2)( √3 – √2)]

⇒ y = [(√3 - √2)²]/[(√3)² – (√2)²]

⇒ y = [3 + 2 - 2√6]/(3 - 2)

⇒ y = (5 - 2√6)

∵ x² = (5 + 2√6)²

⇒ x² = 25 + 24 + 20√6

⇒ x² = 49 + 20√6

∵ y² = (5 - 2√6)²

⇒ y² = 25 + 24 – 20√6

⇒ y² = 49 – 20√6

∴ x² + y² = 49 + 20√6 + 49 – 20√6

⇒ x² + y² = 98