If x=sqrt6+2 and y=sqrt6-2 , then what is the value of (x/y+y/x)²-3 ?

If x=sqrt6+2 and y=sqrt6-2 , then what is the value of…

If $x=\sqrt6+2$ and $y=\sqrt6-2$ , then what is the value of $\left(\frac{x}{y}+\frac{y}{x}\right)^2-3$ ?

Show answer & explanation

Correct answer: B

Explanation

The correct answer is option 2 i.e. 97.

x = √6 + 2 and y = √6 - 2

⇒ x/y = (√6 + 2)/(√6 - 2)

⇒ x/y = [(√6 + 2)(√6 + 2)]/[(√6 - 2)(√6 + 2)]

⇒ x/y = (10 + 4√6)/2 = 5 + 2√6

Similarly, y/x = (√6 - 2)/(√6 + 2)

⇒ y/x = [(√6 - 2)(√6 - 2)]/[(√6 + 2)(√6 - 2)]

⇒ y/x = (10 - 4√6)/2 = 5 - 2√6

Now, (x/y + y/x)² - 3 = [(5 + 2√6) + (5 - 2√6)]² - 3 = 10² - 3 = 100 - 3 = 97

Therefore, the value of the given expression is 97.