A circle has its centre C in the first quadrant. A variable line passing through C meets the positive co-ordinate axes at A and B . The minimum value of OA + OB is 9 and the minimum area of the triangle OAB is 8 , where O is the origin. If the circle passes through the point (1, 1) , then its radius is equal to

Step-by-step solution

Let the centre of the circle be C( , ) in the first quadrant. Let the variable line be x a + y b = 1 . Since it passes through C( , ) , we have a + b = 1 . The sum of intercepts is OA + OB = a + b . We know that the minimum value of a + b subject to a + b = 1 is ( + )^2 . Given that the minimum value is 9 , we have: ( + )^2 = 9 + = 3 The area of triangle OAB is 1 2 ab . Using AM-GM inequality on a + b = 1 : a + b 2 ab 1 2 ab ab 2 ab 4 So, the minimum area is 1 2 (4 ) = 2 . Given that the minimum area is 8 , we have