x2+9y2−4x+6y+4=0
⇒x2−4x+4+9y2+6y+1=1
⇒(x−2)2+(3y+1)2=1
⇒(x−2)2+(y+13)219=1
which is an equation of ellipse having centre at (2,−13).
If P(x,y) is any point on ellipse, then
x=2+cosθ and y=−13+13sinθ
∴4x−9y=4(2+cosθ)−9(−13+13sinθ)
⇒f(θ)=8+4cosθ+3−3sinθ
⇒f(θ)=11+4cosθ−3sinθ
⇒f(θ)max=11+√32+42=16
So maximum value of 4x−9y2=162=8