If `a+ib=sqrt(5+12i)` where a>0, b>0, which of the following is a possible value of `(a^2b^2)` ?

Options

A. 36
B. 9
C. 45
D. 18

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) such that \( a + ib = \sqrt{5 + 12i} \) and then calculate \( a^2b^2 \).

### Step 1: Square both sides
We start with the equation:
\[ a + ib = \sqrt{5 + 12i} \]
Squaring both sides gives:
\[ (a + ib)^2 = 5 + 12i \]

### Step 2: Expand the left side
Expanding the left side:
\[ a^2 + 2a(ib) + (ib)^2 = a^2 + 2abi - b^2 \]
This simplifies to:
\[ a^2 - b^2 + 2abi \]

### Step 3: Set real and imaginary parts equal
Now we equate the real and imaginary parts from both sides:
\[ a^2 - b^2 = 5 \quad \text{(Real part)} \]
\[ 2ab = 12 \quad \text{(Imaginary part)} \]

### Step 4: Solve for \( ab \)
From the imaginary part equation:
\[ 2ab = 12 \implies ab = 6 \]

### Step 5: Substitute \( ab \) into the first equation
Now we have two equations:
1. \( a^2 - b^2 = 5 \)
2. \( ab = 6 \)

We can express \( b \) in terms of \( a \) using \( ab = 6 \):
\[ b = \frac{6}{a} \]

### Step 6: Substitute \( b \) into the first equation
Substituting \( b \) into the first equation:
\[ a^2 - \left(\frac{6}{a}\right)^2 = 5 \]
This simplifies to:
\[ a^2 - \frac{36}{a^2} = 5 \]

### Step 7: Multiply through by \( a^2 \)
To eliminate the fraction, multiply through by \( a^2 \):
\[ a^4 - 36 = 5a^2 \]
Rearranging gives:
\[ a^4 - 5a^2 - 36 = 0 \]

### Step 8: Let \( x = a^2 \)
Let \( x = a^2 \), then we have:
\[ x^2 - 5x - 36 = 0 \]

### Step 9: Solve the quadratic equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-36)}}{2 \cdot 1} \]
\[ x = \frac{5 \pm \sqrt{25 + 144}}{2} \]
\[ x = \frac{5 \pm \sqrt{169}}{2} \]
\[ x = \frac{5 \pm 13}{2} \]
Calculating the two possible values:
\[ x = \frac{18}{2} = 9 \quad \text{or} \quad x = \frac{-8}{2} = -4 \quad (\text{not valid since } x = a^2 > 0) \]

### Step 10: Find \( a^2 \) and \( b^2 \)
Thus, \( a^2 = 9 \). Now we find \( b^2 \):
Using \( ab = 6 \):
\[ b = \frac{6}{a} \implies b^2 = \frac{36}{a^2} = \frac{36}{9} = 4 \]

### Step 11: Calculate \( a^2b^2 \)
Now we can find \( a^2b^2 \):
\[ a^2b^2 = a^2 \cdot b^2 = 9 \cdot 4 = 36 \]

### Final Answer
The possible value of \( a^2b^2 \) is:
\[ \boxed{36} \]