A variable line L passing through the point (3,4) intersects the positive x -axis and positive y -axis at points A and B respectively. Let O be the origin. A rectangle is inscribed in the triangle OAB such that one of its vertices is O and the opposite vertex lies on the line segment AB . The minimum possible value of the maximum area of this rectangle, as L varies, is ______ .

Step-by-step solution

Let the equation of the line L in intercept form be x a + y b = 1 , where a > 0 and b > 0 . Since the line passes through (3,4) , we have: 3 a + 4 b = 1 The area of the right-angled triangle OAB is 1 2 ab . The maximum area of a rectangle inscribed in a triangle and sharing a common vertex (the origin O ) is half the area of the triangle. Thus, the maximum area of the inscribed rectangle is: A_ max = 1 2 ( 1 2 ab ) = ab 4 To find the minimum possible value of A_ max , we need to minimize the product ab . Applying t