Q.
Let ℓ be a line which is normal to the curve y=2x2+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 OPQ is equal to_______
y=2x2+x+2 dxdy=4x+1
Let P be (h,k), then normal at P is y−k=−4h+11(x−h)
This passes through Q(6,4) ∴4−k=−4h+11(6−h) ⇒(4h+1)(4−k)+6−h=0
Also k=2h2+h+2 ∴(4h+1)(4−2h2−h−2)+6+h=0 ⇒4h3−3h2+3h−8=0 ⇒h=1,k=5
Now area of △OPQ will be =21∣∣111016054∣∣=13