If `4x ^(2) + 12 xy - 8x + 9 y ^(2) - 12 y = (ax + by) (ax+ by -4),` then the value of `a ^(2) +b ^(2) ` is

Options

A. 13
B. 25
C. 5
D. 10

Answer

To solve the equation \(4x^2 + 12xy - 8x + 9y^2 - 12y = (ax + by)(ax + by - 4)\), we will first expand the right-hand side and then compare coefficients with the left-hand side.

### Step 1: Expand the right-hand side
The right-hand side can be expanded as follows:
\[ (ax + by)(ax + by - 4) = (ax + by)^2 - 4(ax + by) \]
Now, expanding \((ax + by)^2\):
\[ (ax + by)^2 = a^2x^2 + 2abxy + b^2y^2 \]
Thus,
\[ (ax + by)(ax + by - 4) = a^2x^2 + 2abxy + b^2y^2 - 4(ax + by) \]
Now, expanding \(-4(ax + by)\):
\[ -4(ax + by) = -4ax - 4by \]
Combining these results, we have:
\[ a^2x^2 + 2abxy + b^2y^2 - 4ax - 4by \]

### Step 2: Combine the terms
Putting it all together, we get:
\[ a^2x^2 + 2abxy + b^2y^2 - 4ax - 4by \]

### Step 3: Set the equation equal to the left-hand side
Now, we compare this with the left-hand side:
\[ 4x^2 + 12xy - 8x + 9y^2 - 12y \]
This gives us the following equations by comparing coefficients:
1. \(a^2 = 4\)
2. \(2ab = 12\)
3. \(-4a = -8\)
4. \(-4b = -12\)

### Step 4: Solve for \(a\) and \(b\)
From the third equation:
\[ -4a = -8 \implies a = 2 \]
From the fourth equation:
\[ -4b = -12 \implies b = 3 \]

### Step 5: Calculate \(a^2 + b^2\)
Now, we calculate \(a^2 + b^2\):
\[ a^2 + b^2 = 2^2 + 3^2 = 4 + 9 = 13 \]

### Final Answer
Thus, the value of \(a^2 + b^2\) is \(13\).

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