Let a curve y = f x , x ∈ 0 , ∞ pass through the points P 1 , 3 2 and Q a , 1 2 . If the tangent at any point R ( b , f ( b ) ) to the given curve cuts the y -axis at the point S ( 0 , c ) such that b c = 3 , then P Q 2 is equal to _____.

Step-by-step solution

Given, A curve y = f x ,   x ∈ 0 ,   ∞ pass through the points P 1 ,   3 2 and Q a ,   1 2 , And the tangent at any point R ( b ,   f ( b ) ) to the given curve cuts the y -axis at the point S ( 0 ,   c ) such that b c = 3 , Now tangent to the curve is given by, Y - y = m X - x ,   m = d y d x Now at Y - axis, put X = 0 we get, Y = y - m x or c = y - m x as point S 0 , c given, Now given b c = 3 ⇒ x y - m x = 3 here b = x on the curve ⇒ y - x d y d x = 3 x