The minimum area of a triangle formed by any tangent to the ellipse x^2 16 + y^2 81 =1 and the co-ordinate axes is:

Step-by-step solution

Let (h, k) be the point on ellipse through which tangent is passing. Equation of tangent at (h, k)= x h 16 + y k 81 =1 at y=0, x= 16 h at x=0, y= 81 k Area of AOB = 1 2 ( 16 h ) ( 81 k )= 648 h k A ^2= (648)^2 h^2 k^2 (h, k) must satisfy equation of ellipse aligned & h^2 16 + k^2 81 =1 &h^2= 16 81 (81-k^2 ) aligned Putting value of h^2 in equation (1) A ^2= 81(648)^2 16 k^2 (81-k^2 ) = 81 k^2-k^4 differentiating w.r. to k aligned &2 AA ^ = ( -1 81 k^2-k^4 ) (162 k-4 k^3 ) &2 AA ^ =-2 ~A (81 k-4 k^3 ) & A ^ =-81 k-4