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. $5\sqrt5$
• B. $3\sqrt5$
• C. $2\sqrt5$
• D. $\sqrt5$

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 $\left|2\,\overrightarrow{OP}+3\,\overrightarrow{OQ}\right|$
$=\sqrt{1+4}=\sqrt{5}$