The transformed equation of `5x^(2) + 4xy + 8y^(2) - 12x - 12y =0` when the axes are translated to the point `(1,1//2)` is

Answer

To find the transformed equation of the given equation \(5x^2 + 4xy + 8y^2 - 12x - 12y = 0\) when the axes are translated to the point \((1, \frac{1}{2})\), we will follow these steps:

### Step 1: Define the new coordinates
We need to define the new coordinates \(X\) and \(Y\) in terms of the old coordinates \(x\) and \(y\). The new origin is at the point \((H, K) = (1, \frac{1}{2})\). The transformation equations are:
\[ X = x - H = x - 1 \]
\[ Y = y - K = y - \frac{1}{2} \]

### Step 2: Substitute the new coordinates into the original equation
We will substitute \(x = X + 1\) and \(y = Y + \frac{1}{2}\) into the original equation \(5x^2 + 4xy + 8y^2 - 12x - 12y = 0\).

Substituting:
\[ x = X + 1 \quad \text{and} \quad y = Y + \frac{1}{2} \]

### Step 3: Expand the equation
Now we substitute these values into the equation:
\[ 5(X + 1)^2 + 4(X + 1)(Y + \frac{1}{2}) + 8(Y + \frac{1}{2})^2 - 12(X + 1) - 12(Y + \frac{1}{2}) = 0 \]

### Step 4: Simplify each term
1. Expand \(5(X + 1)^2\):
\[ 5(X^2 + 2X + 1) = 5X^2 + 10X + 5 \]

2. Expand \(4(X + 1)(Y + \frac{1}{2})\):
\[ 4(XY + \frac{1}{2}X + Y + \frac{1}{2}) = 4XY + 2X + 4Y + 2 \]

3. Expand \(8(Y + \frac{1}{2})^2\):
\[ 8(Y^2 + Y + \frac{1}{4}) = 8Y^2 + 8Y + 2 \]

4. Expand \(-12(X + 1)\):
\[ -12X - 12 \]

5. Expand \(-12(Y + \frac{1}{2})\):
\[ -12Y - 6 \]

### Step 5: Combine all terms
Now, we combine all the expanded terms:
\[ 5X^2 + 10X + 5 + 4XY + 2X + 4Y + 2 + 8Y^2 + 8Y + 2 - 12X - 12 - 12Y - 6 = 0 \]

### Step 6: Group similar terms
Now, we will group similar terms:
- \(5X^2\)
- \(8Y^2\)
- \(4XY\)
- Combine \(10X + 2X - 12X = 0\)
- Combine \(4Y + 8Y - 12Y = 0\)
- Combine constants: \(5 + 2 + 2 - 12 - 6 = -9\)

### Final Transformed Equation
Thus, the transformed equation simplifies to:
\[ 5X^2 + 8Y^2 + 4XY - 9 = 0 \]

### Step 7: Replace \(X\) and \(Y\) back to \(x\) and \(y\) (optional)
The final transformed equation can be written as:
\[ 5x^2 + 8y^2 + 4xy - 9 = 0 \]