Question

if x=√2-1/√2+1 and xy=1 then find the value of x²/y-y²/x

pls solve it asap

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Answer 1

Solution :

Given, x = (√2 - 1)/(√2 + 1)

And, xy = 1

We have to find the value of (x²/y) - (y²/x)

⇒ x = (√2 - 1)/(√2 + 1)

We have, xy = 1

⇒ y = 1/{(√2 - 1)/(√2 + 1)}

⇒ y = (√2 + 1)/(√2 - 1)

∴ x - y = (√2 - 1)/(√2 + 1) - (√2 + 1)/(√2 - 1)

= {(√2 - 1)² - (√2 + 1)²}/(√2 + 1)(√2 - 1)

Using identity : a² - b² = (a + b)(a - b)

= {(√2 - 1 + √2 + 1)(√2 - 1 - √2 - 1)}/{(√2)² - 1²}

= {(2√2)(-2)}/(2 - 1)

= -4√2/1

∴ x - y = -4√2

Now we have to find value of :

⇒ (x²/y) - (y²/x) = (x³ - y³)/xy

= {(x - y)³ + 3xy(x - y)}/xy

= {(-4√2)³ + 3 × 1 (-4√2)}/1

[∵ x - y = -4√2 ; xy = 1]

= {-64 × 2√2} - 12√2}/1

= -128√2 - 12√2

= -140√2

Therefore, required value = -140√2