The minimum area of the triangle formed by the tangent  to x2a2+y2b2=1 and the coordinate axes is

Any tangent to the ellipsex2a2+y2b2=1  at P(acos⁡θ,bsin⁡θ) is given byxcos⁡θa+ysin⁡θb=1 It meets the coordinate  axes at A(asec⁡θ,0) and B(0,bcosec⁡θ) . Therefore,  Area of ΔOAB=12×asec⁡θ×bcosec⁡θ or  Δ=absin⁡2θ For area to be minimum, sin⁡2θ should be maximum and we  know that the maximum value of sin⁡2θ is 1 . Therefore, Δmax=ab