`P` and `Q` are two points on the curve `y = 2^(x+2)` in the rectangular cartesian coordinate system such that `bar(OP).barC=-1 and bar(OQ).barC=2`. where `barc` is the unit vector along the positive direction of the x-axis. Then `bar(OQ)-4bar(OP)=` (A) `3i+8j` (B) `4i+6j` (C) `2(3i+4j)` (D) `(4i+3j)`

Answer

To solve the problem step by step, we will first identify the points \( P \) and \( Q \) on the curve \( y = 2^{(x+2)} \) and then find the required expression \( \bar{OQ} - 4\bar{OP} \).

### Step 1: Understand the curve equation
The equation of the curve is given as:
\[ y = 2^{(x+2)} \]
We can rewrite this as:
\[ y = 4 \cdot 2^x \]

### Step 2: Define the unit vector \( \bar{C} \)
The unit vector \( \bar{C} \) along the positive direction of the x-axis is:
\[ \bar{C} = \hat{i} \]

### Step 3: Use the dot product conditions
We are given two conditions:
1. \( \bar{OP} \cdot \bar{C} = -1 \)
2. \( \bar{OQ} \cdot \bar{C} = 2 \)

Since \( \bar{C} = \hat{i} \), we can express \( \bar{OP} \) and \( \bar{OQ} \) in terms of their components:
- Let \( \bar{OP} = x_P \hat{i} + y_P \hat{j} \)
- Let \( \bar{OQ} = x_Q \hat{i} + y_Q \hat{j} \)

### Step 4: Find coordinates of point \( P \)
From the first condition:
\[ \bar{OP} \cdot \bar{C} = x_P = -1 \]
Now substituting \( x_P = -1 \) into the curve equation to find \( y_P \):
\[ y_P = 2^{(-1 + 2)} = 2^1 = 2 \]
Thus, the coordinates of point \( P \) are:
\[ \bar{OP} = -1 \hat{i} + 2 \hat{j} \]

### Step 5: Find coordinates of point \( Q \)
From the second condition:
\[ \bar{OQ} \cdot \bar{C} = x_Q = 2 \]
Now substituting \( x_Q = 2 \) into the curve equation to find \( y_Q \):
\[ y_Q = 2^{(2 + 2)} = 2^4 = 16 \]
Thus, the coordinates of point \( Q \) are:
\[ \bar{OQ} = 2 \hat{i} + 16 \hat{j} \]

### Step 6: Calculate \( \bar{OQ} - 4\bar{OP} \)
Now we need to compute:
\[ \bar{OQ} - 4\bar{OP} = (2 \hat{i} + 16 \hat{j}) - 4(-1 \hat{i} + 2 \hat{j}) \]
Calculating \( 4\bar{OP} \):
\[ 4\bar{OP} = 4(-1 \hat{i} + 2 \hat{j}) = -4 \hat{i} + 8 \hat{j} \]
Now substituting this into our expression:
\[ \bar{OQ} - 4\bar{OP} = (2 \hat{i} + 16 \hat{j}) - (-4 \hat{i} + 8 \hat{j}) = 2 \hat{i} + 16 \hat{j} + 4 \hat{i} - 8 \hat{j} \]
Combining like terms:
\[ = (2 + 4) \hat{i} + (16 - 8) \hat{j} = 6 \hat{i} + 8 \hat{j} \]

### Final Result
Thus, the final result is:
\[ \bar{OQ} - 4\bar{OP} = 6 \hat{i} + 8 \hat{j} \]