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Let x,y in R, then find the maximum and ...

Let `x,y in R`, then find the maximum and minimum values of expression `(x^(2)+y^(2))/(x^(2)+xy+4y^(2))`.

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Verified by Experts

The correct Answer is:
Max. vlaue `=(2)/(5-sqrt(10))`: Min. value `=(2)/(5+sqrt(10))`
Let x=r `cos theta` and y=r `sin theta`
`rArr E=(1)/(4sin^(2)theta+sinthetacos theta+cos^(2)theta)`
`=(2)/(5+sin2theta-3cos 2theta)`
Now, `5-sqrt(10)le5+sin2theta-3cos 2tehta le5+sqrt(10)`
Hence, the maximum value of expression is `(2)/(5-sqrt(10))` and minimum is `(2)/(5+sqrt(10))`
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