If $P$ and $Q$ are two points on the curve $y=2^{x+2}$ such that $\mathbf{O P} \cdot \hat{\mathbf{i}}=-1$…
If $P$ and $Q$ are two points on the curve $y=2^{x+2}$ such that $\mathbf{O P} \cdot \hat{\mathbf{i}}=-1$ and $\mathbf{O Q} \cdot \hat{\mathbf{i}}=2$, then the magnitude of $(\mathbf{Q R}-4 \mathbf{O P})$ is
10
1
5
100
Solution
Let point $P\left(p, 2^{p+2}\right)$ and $Q\left(q, 2^{q+2}\right)$
$
\begin{array}{ll}
\text { Now, } & \mathbf{O P}=P i+2^{p+2} j \\
\text { and } & \mathbf{O Q}=q i+2^{q+2} j
\end{array}
$
According to the question,
$
\begin{aligned}
& \text { OP } \cdot i=-1 \\
& \Rightarrow \quad p=-1 \Rightarrow \mathbf{O P}=-\hat{\mathbf{i}}+2 \hat{\mathbf{j}} \\
& \text { and } \mathbf{O Q} \cdot i=2 \Rightarrow q=2 \\
& \Rightarrow \quad \text { OQ }=2 \hat{\mathbf{i}}+16 \hat{\mathbf{j}} \\
& \text { So, } \quad \mathbf{O Q}-4 \mathbf{O P}=6 \hat{\mathbf{i}}+8 \hat{\mathbf{j}} \\
& \therefore \text { Magnitude of }(\mathbf{O Q}-4 \mathbf{O P})=\sqrt{36+64}=10 \text {. } \\
&
\end{aligned}
$