Question

Let $\ell$ be a line which is normal to the curve $y=2x^2+x+2$ at a point $P$ on the curve. If the point $Q(6,4)$ lies on the line $\ell$ and $O$ is origin, then the area of the triangle $OPQ$ is equal to_______

Answer 1

Answer: 13

$y=2x^2+x+2$

$\frac{dy}{dx}=4x+1$
Let $P$ be $(h,k)$, then normal at $P$ is
$y-k=-\frac{1}{4h+1}(x-h)$
This passes through $Q(6,4)$
$\therefore 4-k=-\frac{1}{4h+1}(6-h)$
$\Rightarrow (4h+1)(4-k)+6-h=0$
Also $k=2h^2+h+2$
$\therefore (4h+1)\left(4-2h^2-h-2\right)+6+h=0$
$\Rightarrow 4h^3-3h^2+3h-8=0$
$\Rightarrow h=1,k=5$
Now area of $△OPQ$ will be $=\frac{1}{2}\left|\begin{array}{ccc}1& 0& 0\\ 1& 1& 5\\ 1& 6& 4\end{array}\right|=13$