Question

Let $P$ and $Q$ be two points in $xy-$plane on the curve $y=x^7-2x^5+5x^3+8x+5$ such that $\overrightarrow{OP}\cdot \hat{i}=2$ and $\overrightarrow{OQ}\cdot \hat{i}=-2$ and the magnitude of $\overrightarrow{OP}+\overrightarrow{OQ}=2M$ $($where $O$ is origin$).$ Then the value of $M$ is

Answer 1

Let $P$ and $Q$ be $(x_1,y_1)$ and $(x_2,y_2)$
$\therefore \overrightarrow{OP}\cdot \hat{i}=x_1=2$
and $\overrightarrow{OQ}\cdot \hat{i}=x_2=-2$
Let $y=f\left(x\right)=x^7-2x^5+5x^3+8x+5$
$\therefore y_1=f\left(x_1\right)=f\left(2\right)$
and $y_2=f\left(x_2\right)=f(-2)$

$\therefore \overrightarrow{OP}+\overrightarrow{OQ}=x_1\hat{i}+y_1\hat{j}+x_2\hat{i}+y_1\hat{j}$
$=(f\left(2\right)+f(-2))\hat{j}$
So, magnitude of $\overrightarrow{OP}+\overrightarrow{OQ}=f\left(2\right)+f(-2)=10$
$\Rightarrow 2M=10\Rightarrow M=5$