Correct Answer - Option 3 : 189/199
Given:
x = 7 + 4√3
xy = 1
Calculation:
xy = 1
⇒ y = 1/x
⇒ y = 1/(7 + 4√3)
⇒ y = 7 - 4√3
(x + y) = 7 + 4√3 + 7 - 4√3
⇒ (x + y) = 14 ----(1)
(x - y) = 7 + 4√3 - 7 + 4√3
⇒ (x - y) = 8√3 ----(2)
We have to find the value of \(\frac{{{x^2}~ -~5xy ~+ \;{y^2}}}{{{x^2}~ + ~5xy ~+ \;{y^2}}}\)
⇒ \(\frac{{{x^2} ~- ~2xy ~+ \;{y^2} - ~3xy}}{{{x^2}~ +~ 2xy ~+ \;{y^2} ~+~ 3xy}}\)
⇒ \(\frac{{{{\left( {x\; - \;y} \right)}^2}\; - \;3xy}}{{{{\left( {x\; + \;y} \right)}^2}\; + \;3xy}}\)
From equation (1) and (2) put the value of (x + y) and (x- y) in the above expression, we get
⇒ {(8√3)2 - 3}/(142 + 3)
⇒ 189/199
∴ The value of \(\frac{{{x^2}~ -~5xy ~+ \;{y^2}}}{{{x^2}~ + ~5xy ~+ \;{y^2}}}\) is 189/199