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Suppose xa n dy are real numbers and tha...

Suppose `xa n dy` are real numbers and that `x^2+9y^2-4x+6y+4=0` . Then the maximum value of `((4x-9y))/2` is__________

Text Solution

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`x^(2)+9y^(2)-4x+6y+4=0`
or `x^(2)-4y+9y^(2)+6y+4=0`
or `(x-2)^(2)+(3y+1)^(2)=1`
or `(x-2)^(2)+({y+(1//3)}^(2))/(1//9)=1`
which is the equation of the ellipse having center at (2,-1//3).
General point on the ellipse is
`P(x,y)-=(2+a cos theta, -(1)/(3)+b sin theta)`
`-=(2 + cos theta, -(1)/(3)+(1)/(3) sin theta)`
`x=2+cos theta and y =-(1)/(3)+(1)/(3) sin theta`
`:. 4x-9y=4(2+cos theta)-9(-(1)/(3)+(1)/(3)sin theta)`
`or f(theta)=8+4cos theta+3-3 sin theta`
`=11+4=16`
`:. f(theta)_("max")=11+5=16`
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