Question

Two points $P$ and $Q$ are lying on the curve $y=log_{2}\left(x + 3\right)$ in $xy$ plane such that $\overset{ \rightarrow }{O P}\cdot \hat{i}=1$ and $\overset{ \rightarrow }{O Q}\cdot \hat{j}=3$ , then the value of $\left|\overset{ \rightarrow }{O Q} - 2 \overset{ \rightarrow }{O P}\right|$ 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 $Q\left(x_{2}, \log _{2}\left(x_{2}+3\right)\right)$
$\because \overrightarrow{O P} \cdot \hat{i}=1 \Rightarrow x_{1}=1 \Rightarrow P(1,2)$
$\because \overrightarrow{O Q} \cdot \hat{j}=3 \Rightarrow \log _{2}\left(x_{2}+3\right)=3 \Rightarrow x_{2}=5 \Rightarrow Q(5,3)$
$\overrightarrow{O Q}-2 \overrightarrow{O P}|=| 5 \hat{i}+3 \hat{j}-2(\hat{i}+2 \hat{j}) \mid$
$=|3 \hat{i}-\hat{j}|=\sqrt{10}$