Find the point to which the axes are to be translated to eliminate x and y terms (remove first degree terms) in the equation `2x^(2)+4xy+5y^(2)-4x-22y+7 = 0.`

Answer

To eliminate the first-degree terms in the equation \(2x^2 + 4xy + 5y^2 - 4x - 22y + 7 = 0\), we will follow these steps:

### Step 1: Set Up the Translation of Axes
Let the new origin be at the point \((H, K)\). We will use the transformations:
\[ X = x - H \quad \text{and} \quad Y = y - K \]
where \(X\) and \(Y\) are the coordinates with respect to the new origin.

### Step 2: Substitute the Transformations into the Equation
We substitute \(x = X + H\) and \(y = Y + K\) into the original equation:
\[ 2(X + H)^2 + 4(X + H)(Y + K) + 5(Y + K)^2 - 4(X + H) - 22(Y + K) + 7 = 0 \]

### Step 3: Expand the Equation
Now we will expand each term in the equation:
1. \(2(X + H)^2 = 2(X^2 + 2HX + H^2) = 2X^2 + 4HX + 2H^2\)
2. \(4(X + H)(Y + K) = 4(XY + XK + HY + HK) = 4XY + 4XK + 4HY + 4HK\)
3. \(5(Y + K)^2 = 5(Y^2 + 2KY + K^2) = 5Y^2 + 10KY + 5K^2\)
4. \(-4(X + H) = -4X - 4H\)
5. \(-22(Y + K) = -22Y - 22K\)
6. The constant term remains \(+7\).

Combining all these, we have:
\[ 2X^2 + 4XY + 5Y^2 + (4H + 4K - 4)X + (4H + 10K - 22)Y + (2H^2 + 4HK + 5K^2 + 7) = 0 \]

### Step 4: Collect the First-Degree Terms
The first-degree terms in the equation are:
- Coefficient of \(X\): \(4H + 4K - 4\)
- Coefficient of \(Y\): \(4H + 10K - 22\)

### Step 5: Set the Coefficients to Zero
To eliminate the first-degree terms, we set the coefficients of \(X\) and \(Y\) to zero:
1. \(4H + 4K - 4 = 0\) (Equation 1)
2. \(4H + 10K - 22 = 0\) (Equation 2)

### Step 6: Solve the System of Equations
From Equation 1:
\[ 4H + 4K = 4 \implies H + K = 1 \quad \text{(divide by 4)} \]
From Equation 2:
\[ 4H + 10K = 22 \implies 2H + 5K = 11 \quad \text{(divide by 2)} \]

Now we have the system:
1. \(H + K = 1\)
2. \(2H + 5K = 11\)

### Step 7: Substitute and Solve for \(H\) and \(K\)
From \(H + K = 1\), we can express \(H\) in terms of \(K\):
\[ H = 1 - K \]
Substituting into the second equation:
\[ 2(1 - K) + 5K = 11 \]
\[ 2 - 2K + 5K = 11 \]
\[ 3K = 9 \implies K = 3 \]
Substituting \(K = 3\) back into \(H + K = 1\):
\[ H + 3 = 1 \implies H = 1 - 3 = -2 \]

### Step 8: Conclusion
The point to which the axes should be translated is \((H, K) = (-2, 3)\).