If `4x^(2) + 12xy -8x+9y^(2) -12y = (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 expand the right-hand side and then compare coefficients with the left-hand side.

### Step 1: Expand the Right-Hand Side
We start by expanding the expression \((ax + by)(ax + by - 4)\).

\[ (ax + by)(ax + by - 4) = (ax + by)(ax + by) - 4(ax + by) \]

Now, we expand \((ax + by)(ax + by)\):

\[ (ax + by)(ax + by) = a^2x^2 + 2abxy + b^2y^2 \]

Next, we expand \(-4(ax + by)\):

\[ -4(ax + by) = -4ax - 4by \]

Combining these two expansions gives us:

\[ a^2x^2 + 2abxy + b^2y^2 - 4ax - 4by \]

### Step 2: Combine Like Terms
Now, we can write the full expansion:

\[ a^2x^2 + 2abxy + b^2y^2 - 4ax - 4by \]

### Step 3: Set Up the Equation
We now set this equal to the left-hand side of the original equation:

\[ 4x^2 + 12xy - 8x + 9y^2 - 12y = a^2x^2 + 2abxy + b^2y^2 - 4ax - 4by \]

### Step 4: Compare Coefficients
Now we compare the coefficients of \(x^2\), \(xy\), \(x\), \(y^2\), and \(y\) from both sides:

1. Coefficient of \(x^2\):
\[ a^2 = 4 \]
Thus, \(a = 2\) or \(a = -2\).

2. Coefficient of \(xy\):
\[ 2ab = 12 \]

3. Coefficient of \(x\):
\[ -4a = -8 \implies 4a = 8 \implies a = 2 \]

4. Coefficient of \(y^2\):
\[ b^2 = 9 \implies b = 3 \text{ or } b = -3 \]

5. Coefficient of \(y\):
\[ -4b = -12 \implies 4b = 12 \implies b = 3 \]

### Step 5: Calculate \(a^2 + b^2\)
Now that we have \(a = 2\) and \(b = 3\), we can find \(a^2 + b^2\):

\[ a^2 + b^2 = 2^2 + 3^2 = 4 + 9 = 13 \]

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

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