The point to which the origin should be shifted in order to eliminate x and y terms in the equation `4x^(2) + 9y^(2) - 8x + 36y +4=0` is

Answer

To find the point to which the origin should be shifted in order to eliminate the x and y terms in the equation \(4x^2 + 9y^2 - 8x + 36y + 4 = 0\), we can follow these steps:

### Step 1: Set Up the Translation of Axes
We will use the translation of axes formula:
\[ X = x - h \quad \text{and} \quad Y = y - k \]
where \((h, k)\) is the new origin we need to find.

### Step 2: Substitute into the Equation
Substituting \(x\) and \(y\) in terms of \(X\) and \(Y\) into the given equation:
\[ 4(x + h)^2 + 9(y + k)^2 - 8(x + h) + 36(y + k) + 4 = 0 \]

### Step 3: Expand the Equation
Now, we will expand the equation:
\[ 4(x^2 + 2xh + h^2) + 9(y^2 + 2yk + k^2) - 8(x + h) + 36(y + k) + 4 = 0 \]
This simplifies to:
\[ 4x^2 + 8xh + 4h^2 + 9y^2 + 18yk + 9k^2 - 8x - 8h + 36y + 36k + 4 = 0 \]

### Step 4: Collect Like Terms
Now, we collect the terms involving \(x\) and \(y\):
\[ (4x^2 + 9y^2) + (8h - 8)x + (18k + 36)y + (4h^2 + 9k^2 - 8h + 36k + 4) = 0 \]

### Step 5: Set Coefficients of \(x\) and \(y\) to Zero
To eliminate the \(x\) and \(y\) terms, we set their coefficients to zero:
1. For \(x\):
\[ 8h - 8 = 0 \implies h = 1 \]
2. For \(y\):
\[ 18k + 36 = 0 \implies 18k = -36 \implies k = -2 \]

### Step 6: Determine the New Origin
Thus, the new origin to which the point should be shifted is:
\[ (h, k) = (1, -2) \]

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

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