Let a smooth curve y = f x be such that the slope of the tangent at any point x , y on it is directly proportional to - y x . If the curve passes through the points 1 , 2 and 8 , 1 , then y 1 8 is equal to

Step-by-step solution

Given, d y d x ∝ - y x ⇒ d y d x = - K y x (where K is proportionality constant) Now integrating both side, ⇒ ∫ d y y = - K ∫ d x x ⇒ ln y = - K ln x + C If the above equation satisfy 1 , 2 ⇒ ln 2 = - K × 0 + C ⇒ C = ln 2 So, ln y = - K ln x + ln 2 Now it also passes through 8 , 1 ⇒ ln 1 = - K ln 8 + ln 2 ⇒ K = 1 3 So, equation becomes ln y = - 1 3 ln x + ln 2 So, at x = 1 8 ⇒ ln y = - 1 3 ln 1 8 + ln 2 = 2 ln 2 ⇒ y = 4