The coordinates of the point where origin is shifted is (-1,2) so that the equation`2x^(2)+y^(2)-4x+4y=0` become?

Options

A. `X^(2)+2Y^(2)=6`
B. `2X^(2)+Y^(2)=6`
C. `2X^(2)+Y^(2)=4`
D. `X^(2)+2Y^(2)=4`

Explanation

To solve the problem of shifting the origin to the point (-1, 2) for the equation \(2x^2 + y^2 - 4x + 4y = 0\), we will follow these steps:

### Step 1: Define the new coordinates
When we shift the origin to the point (-1, 2), we need to define the new coordinates \(X\) and \(Y\) in terms of the old coordinates \(x\) and \(y\):
\[ X = x + 1 \quad \text{and} \quad Y = y - 2 \]
This means:
\[ x = X - 1 \quad \text{and} \quad y = Y + 2 \]

### Step 2: Substitute the new coordinates into the equation
Now, we substitute \(x\) and \(y\) in the original equation \(2x^2 + y^2 - 4x + 4y = 0\):
\[ 2(X - 1)^2 + (Y + 2)^2 - 4(X - 1) + 4(Y + 2) = 0 \]

### Step 3: Expand the equation
Now, we will expand each term:
1. \(2(X - 1)^2 = 2(X^2 - 2X + 1) = 2X^2 - 4X + 2\)
2. \((Y + 2)^2 = Y^2 + 4Y + 4\)
3. \(-4(X - 1) = -4X + 4\)
4. \(4(Y + 2) = 4Y + 8\)

Now substituting these back into the equation:
\[ 2X^2 - 4X + 2 + Y^2 + 4Y + 4 - 4X + 4 + 4Y + 8 = 0 \]

### Step 4: Combine like terms
Now, we will combine all the terms:
\[ 2X^2 + Y^2 + (-4X - 4X) + (4Y + 4Y) + (2 + 4 + 4 + 8) = 0 \]
This simplifies to:
\[ 2X^2 + Y^2 - 8X + 8Y + 18 = 0 \]

### Step 5: Rearranging the equation
Now, we can rearrange the equation to isolate the constant term:
\[ 2X^2 + Y^2 - 8X + 8Y = -18 \]

### Step 6: Final form of the equation
To express the equation in a standard form, we can move the constant to the other side:
\[ 2X^2 + Y^2 = 18 \]

### Step 7: Divide by 6 to simplify
To make it more standard, we can divide the entire equation by 6:
\[ \frac{2X^2}{6} + \frac{Y^2}{6} = 1 \implies \frac{X^2}{3} + \frac{Y^2}{6} = 1 \]

Thus, the final equation after shifting the origin is:
\[ 2X^2 + Y^2 = 18 \]