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 :

Question

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 :, A circle of radius = 2, A hyperbola with each semi-axis = 2 ., A hyperbola with each semi-axis = 2, A circle of radius = 2

MARKS App · Accepted Answer

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.

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