Question

If \(P,Q\) are two points on the curve \(y=2^{x+2}\) in the rectangular Cartesian coordinate system such that \(\overrightarrow{OP}\cdot \vec{i}=-1,\;\overrightarrow{OQ}\cdot \vec{i}=2\), then \(\overrightarrow{OQ}-4\overrightarrow{OP}=\)

• A. \(3\vec{i}+8\vec{j}\)
• B. \(4\vec{i}+6\vec{j}\)
• C. \(6\vec{i}+8\vec{j}\)
• D. \(4\vec{i}+3\vec{j}\)

Hint:

For a position vector \(\overrightarrow{OP}=x\vec{i}+y\vec{j}\), the value of \(\overrightarrow{OP}\cdot \vec{i}\) gives the \(x\)-coordinate of the point \(P\).

Answer 1

The Correct Option is C

Step 1: Understand the given curve.
The given curve is
\[ y=2^{x+2} \] For any point on this curve, if the \(x\)-coordinate is known, then the \(y\)-coordinate can be found using this equation.

Step 2: Find the coordinates of point \(P\).
Given,
\[ \overrightarrow{OP}\cdot \vec{i}=-1 \] This means the \(x\)-coordinate of \(P\) is \(-1\).
So,
\[ x=-1 \] Now, using \(y=2^{x+2}\),
\[ y=2^{-1+2}=2^1=2 \] Therefore,
\[ P=(-1,2) \] Hence,
\[ \overrightarrow{OP}=-\vec{i}+2\vec{j} \]

Step 3: Find the coordinates of point \(Q\).
Given,
\[ \overrightarrow{OQ}\cdot \vec{i}=2 \] This means the \(x\)-coordinate of \(Q\) is \(2\).
So,
\[ x=2 \] Now, using \(y=2^{x+2}\),
\[ y=2^{2+2}=2^4=16 \] Therefore,
\[ Q=(2,16) \] Hence,
\[ \overrightarrow{OQ}=2\vec{i}+16\vec{j} \]

Step 4: Calculate \(\overrightarrow{OQ}-4\overrightarrow{OP}\).
We have,
\[ \overrightarrow{OQ}=2\vec{i}+16\vec{j} \] and
\[ \overrightarrow{OP}=-\vec{i}+2\vec{j} \] Therefore,
\[ \overrightarrow{OQ}-4\overrightarrow{OP} = (2\vec{i}+16\vec{j})-4(-\vec{i}+2\vec{j}) \] \[ =2\vec{i}+16\vec{j}+4\vec{i}-8\vec{j} \] \[ =6\vec{i}+8\vec{j} \]

Step 5: Final conclusion.
Hence,
\[ \boxed{6\vec{i}+8\vec{j}} \]