If the tangent to the curve y = x 3 - x 2 + x at the point a , b is also tangent to the curve y = 5 x 2 + 2 x - 25 at the point 2 , - 1 , then 2 a + 9 b is equal to ______.

Step-by-step solution

Given, Curve y = 5 x 2 + 2 x - 25 Now differentiating to find slope at point P 2 , - 1 we get, y ' = 10 x + 2 ⇒ y P ' = 22 Now tangent to curve at P will be, y + 1 = 22 x - 2 ⇒ y = 22 x - 45 Now the tangent is also tangent to the curve y = x 3 - x 2 + x and passing through a , b so plottnig the diagram we get, Now differentiating the curve y = x 3 - x 2 + x we get, d y d x = 3 x 2 - 2 x + 1 d y d x Q = a , b = 3 a 2 - 2 a + 1 So, equating with tangent slope we get, 3 a 2 - 2 a + 1 = 22 ⇒ 3