Question

Two points $P$ and $Q$ are lying on the curve $y=lo{g}_2\left(x+3\right)$ in $xy$ plane such that $\overrightarrow{\mathrm{OP}}\cdot \hat{i}=1$ and $\overrightarrow{\mathrm{OQ}}\cdot \hat{j}=3$ , then the value of $\left|\overrightarrow{\mathrm{OQ}}-2\overrightarrow{\mathrm{OP}}\right|$ is

(where, $O$ is the origin)

• A. $\sqrt{6}$
• B. $\sqrt{7}$
• C. $\sqrt{8}$
• D. $\sqrt{10}$

Answer 1

Answer: (D)

Let $P\left(x_1,{\log }_2\left(x_1+3\right)\right)$ and $Q\left(x_2,{\log }_2\left(x_2+3\right)\right)$
$\because \overrightarrow{OP}\cdot \hat{i}=1\Rightarrow x_1=1\Rightarrow P(1,2)$
$\because \overrightarrow{OQ}\cdot \hat{j}=3\Rightarrow {\log }_2\left(x_2+3\right)=3\Rightarrow x_2=5\Rightarrow Q(5,3)$
$\overrightarrow{OQ}-2\overrightarrow{OP}|=|5\hat{i}+3\hat{j}-2(\hat{i}+2\hat{j})\mid$
$=|3\hat{i}-\hat{j}|=\sqrt{10}$