Let a and b be any two numbers satisfying 1 a 2 + 1 b 2 = 1 4 . Then, the foot of perpendicular from the origin on the variable line x a + y b = 1 lies on :

Step-by-step solution

The given line is b x + a y = a b . Foot of perpendicular from origin x b = y a = - - a b a 2 + b 2 x = a b 2 a 2 + b 2 ,   y = a 2 b a 2 + b 2 x 2 + y 2 = a 2 b 2 a 2 + b 2 a 2 + b 2 2 x 2 + y 2 = a 2 b 2 a 2 + b 2 given 1 a 2 + 1 b 2 = 1 4   a 2 + b 2 a 2 b 2 = 1 4 ⇒   x 2 + y 2 = 4 , which is equation of circle.