When the origin is shifted to a suitable point, the equation `2x^2+y^2-4x+4y=0` transformed as 2x^2+y^2-8x +8y+ 18=0`. The point to which origin was shifted is

Options

A. (1,2)
B. (1,-2)
C. (-1,2)
D. (-1,-2)

Answer

To solve the problem, we need to find the point to which the origin was shifted, given the transformation of the equation. Here’s a step-by-step solution:

### Step 1: Understand the transformation
We start with the original equation:
\[ 2x^2 + y^2 - 4x + 4y = 0 \]
and it transforms to:
\[ 2X^2 + Y^2 - 8X + 8Y + 18 = 0 \]
where \( X \) and \( Y \) are the new coordinates after shifting the origin.

### Step 2: Define the shift in origin
Let the origin be shifted to a point \( (\alpha, \beta) \). Therefore, we have:
\[ x = X + \alpha \]
\[ y = Y + \beta \]

### Step 3: Substitute the new coordinates into the original equation
Substituting \( x \) and \( y \) into the original equation gives:
\[ 2(X + \alpha)^2 + (Y + \beta)^2 - 4(X + \alpha) + 4(Y + \beta) = 0 \]

### Step 4: Expand the equation
Expanding the equation:
\[ 2(X^2 + 2\alpha X + \alpha^2) + (Y^2 + 2\beta Y + \beta^2) - 4X - 4\alpha + 4Y + 4\beta = 0 \]
This simplifies to:
\[ 2X^2 + 4\alpha X + 2\alpha^2 + Y^2 + 2\beta Y + \beta^2 - 4X - 4\alpha + 4Y + 4\beta = 0 \]

### Step 5: Combine like terms
Combining like terms, we have:
\[ 2X^2 + Y^2 + (4\alpha - 4)X + (2\beta + 4)Y + (2\alpha^2 - 4\alpha + \beta^2 + 4\beta) = 0 \]

### Step 6: Compare coefficients
Now, we compare this equation with the transformed equation:
\[ 2X^2 + Y^2 - 8X + 8Y + 18 = 0 \]

From this, we can set up the following equations by comparing coefficients:
1. \( 4\alpha - 4 = -8 \)
2. \( 2\beta + 4 = 8 \)
3. \( 2\alpha^2 - 4\alpha + \beta^2 + 4\beta = 18 \)

### Step 7: Solve for \( \alpha \)
From the first equation:
\[ 4\alpha - 4 = -8 \implies 4\alpha = -4 \implies \alpha = -1 \]

### Step 8: Solve for \( \beta \)
From the second equation:
\[ 2\beta + 4 = 8 \implies 2\beta = 4 \implies \beta = 2 \]

### Step 9: Conclusion
Thus, the point to which the origin was shifted is:
\[ (\alpha, \beta) = (-1, 2) \]

### Final Answer
The point to which the origin was shifted is \( (-1, 2) \).

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