What will be the value of p if the eqution of straight line 2x + 5y = 4 gets changed to 2x + 5y = p after shifting the origin at (3, 3)?

Options

A. 16
B. -17
C. 12
D. 10

Answer

To find the value of \( p \) when the equation of the straight line \( 2x + 5y = 4 \) is transformed after shifting the origin to the point \( (3, 3) \), we can follow these steps:

### Step-by-step Solution:

1. **Identify the Original Equation**:
The original equation of the line is given as:
\[ 2x + 5y = 4 \]

2. **Shift the Origin**:
When the origin is shifted from \( (0, 0) \) to \( (3, 3) \), we need to replace \( x \) and \( y \) in the equation with \( x' \) and \( y' \) where:
\[ x' = x - 3 \quad \text{and} \quad y' = y - 3 \]
Thus, we can express \( x \) and \( y \) in terms of \( x' \) and \( y' \):
\[ x = x' + 3 \quad \text{and} \quad y = y' + 3 \]

3. **Substitute into the Original Equation**:
Substitute \( x \) and \( y \) into the original equation:
\[ 2(x' + 3) + 5(y' + 3) = 4 \]

4. **Expand the Equation**:
Expanding the equation gives:
\[ 2x' + 6 + 5y' + 15 = 4 \]

5. **Combine Like Terms**:
Combine the constant terms:
\[ 2x' + 5y' + 21 = 4 \]

6. **Rearrange the Equation**:
Rearranging the equation to isolate the terms involving \( x' \) and \( y' \):
\[ 2x' + 5y' = 4 - 21 \]
\[ 2x' + 5y' = -17 \]

7. **Identify the New Value of \( p \)**:
The new equation after shifting the origin is:
\[ 2x' + 5y' = p \]
From our rearranged equation, we see that:
\[ p = -17 \]

### Final Answer:
Thus, the value of \( p \) is:
\[ \boxed{-17} \]