If origin is shifted to `(-2,3)` then transformed equation of curve `x^2 +2y-3=0` w.r.t. to `(0,0)` is

Options

A. `x^(2)-4x +2y +4 = 0`
B. `x^(2) - 4x - 2y - 5 = 0`
C. `x^(2) +4x +2y - 5 = 0`
D. None of these

Explanation

To find the transformed equation of the curve \( x^2 + 2y - 3 = 0 \) when the origin is shifted to \((-2, 3)\), we will follow these steps:

### Step 1: Understand the Shift of Origin
When the origin is shifted from \((0, 0)\) to \((-2, 3)\), we need to express the new coordinates in terms of the old coordinates. The new coordinates \( (x', y') \) can be expressed as:
\[ x' = x + 2 \]
\[ y' = y - 3 \]

### Step 2: Substitute the New Coordinates into the Original Equation
The original equation is:
\[ x^2 + 2y - 3 = 0 \]
We will substitute \( x = x' - 2 \) and \( y = y' + 3 \) into this equation.

### Step 3: Substitute and Simplify
Substituting \( x \) and \( y \) into the equation:
\[ (x' - 2)^2 + 2(y' + 3) - 3 = 0 \]
Now, we will expand and simplify this expression.

1. Expand \( (x' - 2)^2 \):
\[ (x' - 2)^2 = x'^2 - 4x' + 4 \]

2. Expand \( 2(y' + 3) \):
\[ 2(y' + 3) = 2y' + 6 \]

3. Combine everything:
\[ x'^2 - 4x' + 4 + 2y' + 6 - 3 = 0 \]
This simplifies to:
\[ x'^2 - 4x' + 2y' + 7 = 0 \]

### Step 4: Write the Final Transformed Equation
The transformed equation with respect to the new origin \((-2, 3)\) is:
\[ x'^2 - 4x' + 2y' + 7 = 0 \]