The tangent to the ellipse 9 x^2+16 y^2=288 making equal intercepts on the co-ordinate axes intersects the X -axis and the Y -axis in the points A and B respectively. Then A( O A B)= (where O is origin)

Step-by-step solution

A line tangent to the ellipse x^2 32 + y^2 18 = 1 and making equal intercepts on the coordinate axes has equation x + y = k , where k is the intercept length. The condition for tangency c^2 = a^2m^2 + b^2 gives k^2 = (32)(-1)^2 + 18 = 50 . The triangle OAB is formed by the line with intercept k intersecting the axes at A(k, 0) and B(0, k) , and the origin O(0, 0) . The area is given by 1 2 |k| |k| = 1 2 k^2 = 1 2 50 = 25 square units.