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If `x ,y in R ,` satisfies the equation `((x-4)^2)/4+(y^2)/9=1` , then the difference between the largest and the smallest valus of the expression `(x^2)/4+(y^2)/9` is_____

Text Solution

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Let `x-4=2 cos theta, i.e.,x=2 cos theta+4 and y=3 sin theta`. Now,
`E=(x^(2))/(4)(y^(2))/(9)=((2 cos theta+4)^(2))/(4)+sin^(2)theta`
`(4cos^(2)theta+16+cos theta+4 sin^(2)theta)/(4)`
`(20+cos theta)/(4)=5+4 cos theta`
Hence ` E_("max")-E_("min")=9-1=8`
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