Let a variable line passing through the centre of the circle x 2 + y 2 − 16 x − 4 y = 0 , meet the positive co-ordinate axes at the point A and B . Then the minimum value of O A + O B , where O is the origin, is equal to

Step-by-step solution

Given: A line passing through the centre of circle x 2 + y 2 - 16 x - 4 y = 0 which is 8 , 2 intersects positive x & y axis at A & B respectively, So, let the line be x a + y b = 1 Now, given line passes through the centre, So, 8 a + 2 b = 1 Now, using A . M ≥ H . M we get, a 2 + a 2 + a 2 + a 2 + b 1 + b 1 6 ≥ 6 2 a + 2 a + 2 a + 2 a + 1 b + 1 b ⇒ 2 a + 2 b 6 ≥ 6 1 ⇒ 2 a + 2 b ≥ 36 ⇒ a + b ≥ 18 ⇒ O A + O B ≥ 18 Hence, the minimum value of O A + O B = 18