If the equation of the normal to the curve y = x - a x + b x - 2 at the point 1 , - 3 is x - 4 y = 13 then the value of a + b is equal to ______

Step-by-step solution

Given curve is y = x - a x + b x - 2       . . . 1 At point 1 , - 3 , we have - 3 = 1 - a 1 + b 1 - 2 ⇒ 3 1 + b = 1 - a ⇒ 1 - a = 3 + 3 b Differentiating 1 both sides w.r.t. x , we get d y d x = ( x + b ) ( x - 2 ) × ( 1 ) - ( x - a ) ( 2 x + b - 2 ) ( x + b ) 2 ( x - 2 ) 2 At ( 1 , - 3 ) slope of normal is 1 4 , hence d x d y = - 1 4 ⇒ d y d x 1 , - 3 = - 4 , ⇒ - 4 = ( 1 + b ) ( - 1 ) - ( 1 - a ) b ( 1 + b ) 2 ( - 1 ) 2 ⇒ - 4 = ( 1 + b ) ( - 1 ) - 3 ( b + 1 ) b ( 1