A straight line L passes through the point of intersection of the lines x - y + 1 = 0 and 2x + y - 7 = 0 . If L intersects the positive x-axis at A(a, 0) and the positive y-axis at B(0, b) , then the minimum area of the triangle OAB (where O is the origin) is ....

Step-by-step solution

Solving the equations x - y + 1 = 0 and 2x + y - 7 = 0 , we get the point of intersection as (2, 3) . Let the equation of the line L be x a + y b = 1 . Since it passes through (2, 3) , we have: 2 a + 3 b = 1 Since L intersects the positive x-axis and positive y-axis, a > 0 and b > 0 . Applying the AM-GM inequality on 2 a and 3 b : 2 a + 3 b 2 2 a 3 b 1 2 6 ab Squaring both sides: 1 4 6 ab ab 24 The area of the triangle OAB is given by = 1 2 ab . 1 2 24 = 12 The minimum area of the triangle is 12 square units.