Transform the equation `x^(2)-3xy+11x-12y+36=0` to parallel axes through the point (-4, 1) becomes `ax^(2)+bxy+1=0` then `b^(2)-a=`

Options

A. `(1)/(4)`
B. `(1)/(16)`
C. `(1)/(64)`
D. `(1)/(256)`

Answer

To solve the equation \( x^2 - 3xy + 11x - 12y + 36 = 0 \) and transform it to the form \( ax^2 + bxy + 1 = 0 \) through the point (-4, 1), we will follow these steps:

### Step 1: Substitute the new coordinates
We start by letting:
\[ X = x + 4 \quad \text{and} \quad Y = y - 1 \]
This means:
\[ x = X - 4 \quad \text{and} \quad y = Y + 1 \]

### Step 2: Substitute \( x \) and \( y \) into the original equation
Substituting \( x \) and \( y \) into the original equation:
\[ (X - 4)^2 - 3(X - 4)(Y + 1) + 11(X - 4) - 12(Y + 1) + 36 = 0 \]

### Step 3: Expand the equation
Now we expand each term:
1. \( (X - 4)^2 = X^2 - 8X + 16 \)
2. \( -3(X - 4)(Y + 1) = -3(XY + X - 4Y - 4) = -3XY - 3X + 12Y + 12 \)
3. \( 11(X - 4) = 11X - 44 \)
4. \( -12(Y + 1) = -12Y - 12 \)

Combining these, we get:
\[ X^2 - 8X + 16 - 3XY - 3X + 12Y + 12 + 11X - 44 - 12Y - 12 + 36 = 0 \]

### Step 4: Simplify the equation
Now, combine like terms:
\[ X^2 + (-8X - 3X + 11X) + (16 + 12 - 44 - 12 + 36) - 3XY + (12Y - 12Y) = 0 \]
This simplifies to:
\[ X^2 + 0X + 8 - 3XY = 0 \]
or:
\[ X^2 - 3XY + 8 = 0 \]

### Step 5: Compare with the desired form
We need to express this in the form \( ax^2 + bxy + 1 = 0 \). To do this, we can divide the entire equation by 8:
\[ \frac{1}{8}X^2 - \frac{3}{8}XY + 1 = 0 \]
From this, we identify:
\[ a = \frac{1}{8}, \quad b = -\frac{3}{8} \]

### Step 6: Calculate \( b^2 - a \)
Now, we compute:
\[ b^2 - a = \left(-\frac{3}{8}\right)^2 - \frac{1}{8} = \frac{9}{64} - \frac{8}{64} = \frac{1}{64} \]

### Final Answer
Thus, the value of \( b^2 - a \) is:
\[ \frac{1}{64} \]