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Question

If a tangent to the ellipse x2a2+y2b2=1 having slope 2 is normal to the circle x2+y2+4x+1=0, then the maximum value of ab is

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Solution

Equation of tangent is y=2x±4a2+b2

If this tangent is normal to the circle x2+y2+4x+1=0, then it will pass through the centre (2,0) of the circle.
0=4±4a2+b24a2+b2=16

Using A.M.G.M. inequality, we get
4a2+b224a2×b2ab4

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