Let l be a line which is normal to the curve y = 2 x 2 + x + 2 at a point P on the curve. If the point Q 6 , 4 lies on the line l and O is origin, then the area of the triangle O P Q is equal to _____.

Step-by-step solution

The given curve is y = 2 x 2 + x + 2 d y d x = 4 x + 1 Let P be h , k , then normal at P is y - k = - 1 4 h + 1 x - h Since this normal passes through Q 6 , 4 ∴   4 - k = - 1 4 h + 1 6 - h ⇒ 4 h + 1 4 - k + 6 - h = 0 Also k = 2 h 2 + h + 2 ∴   4 h + 1 4 - 2 h 2 - h - 2 + 6 + h = 0 ⇒ 4 h 3 - 3 h 2 + 3 h - 8 = 0 ⇒ h = 1 , k = 5 So P ≡ 1 , 5 Now area of ∆ O P Q will be = 1 2 1 0 0 1 1 5 1 6 4 = 13