if a tangent of slope2of the ellipse x2a2+y2b2=1 is normal to the circle x2+y2+4x+1=0 then the maximum value of ab is_________

The equation of tangent is y=2x±4a2+b2This is normal to the circle x2+y2+4x+1=0This tangent passes through (-2,0). Therefore0=−4±4a2⋅b2  or 4a2+b2=16usingAM≥GM we get4a2+b22≥4a2+b2  or  ab≤4