Question

If P and Q be two given points on the curve $y=x+\frac{1}{x}$ such that $\overrightarrow{OP}.\hat{i}=1$ and $\overrightarrow{OQ}.\hat{i}=-1$ where $\hat{i}$ is a unit vector along the x-axis, then the length of vector $2\overrightarrow{OP}+3\overrightarrow{OQ}$ is

• A. $5\sqrt{5}$
• B. $3\sqrt{5}$
• C. $2\sqrt{5}$
• D. $\sqrt{5}$

Answer 1

Answer: (D)

Let $P(x_1,y_1)$, $O(x_2,y_2)$ be two points on the curve
$y=x+\frac{1}{x}$
Let $\vec{j}$ be a unit vector along $y$-axis.
Then $\overrightarrow{OP}=x_1\hat{i}+y_1\hat{j},\overrightarrow{OQ}=x_2\hat{i}+y_2\hat{j}$
Since $\overrightarrow{OP}.\hat{i}=-1$ and $\overrightarrow{OQ}.\hat{j}=-1$
$\therefore x_1=1$ and $x_2=-1$
$\Rightarrow y_1=2$ and $y_2=-2$
$\therefore \overrightarrow{OP}=\hat{i}+2\hat{j}$ ;
$\overrightarrow{OQ}=-\hat{i}-2\hat{j}$
$\therefore 2\overrightarrow{OP}+3\overrightarrow{OQ}=2\hat{i}+4\hat{j}-3\hat{i}-6\hat{j}$
$=-\hat{i}-2\hat{j}$
Hence $\left|2\overrightarrow{OP}+3\overrightarrow{OQ}\right|$
$=\sqrt{1+4}=\sqrt{5}$