If `1+4sqrt(3)i=(1+ib)^(2)`, prove that:
`a^(2)-b^(2)=1 and ab=2sqrt(3)`.

Answer

To solve the equation \(1 + 4\sqrt{3}i = (1 + ib)^2\) and prove that \(a^2 - b^2 = 1\) and \(ab = 2\sqrt{3}\), we will follow these steps:

### Step 1: Expand the right-hand side
We start with the expression \((1 + ib)^2\):
\[ (1 + ib)^2 = 1^2 + 2(1)(ib) + (ib)^2 \]
\[ = 1 + 2ib + i^2b^2 \]
Since \(i^2 = -1\), we can substitute:
\[ = 1 + 2ib - b^2 \]
Thus, we can rewrite it as:
\[ = (1 - b^2) + 2ib \]

### Step 2: Equate real and imaginary parts
Now we equate the real and imaginary parts of both sides of the equation:
\[ 1 + 4\sqrt{3}i = (1 - b^2) + 2ib \]
From this, we can set up two equations:
1. Real part: \(1 - b^2 = 1\)
2. Imaginary part: \(2b = 4\sqrt{3}\)

### Step 3: Solve the real part equation
From the real part equation:
\[ 1 - b^2 = 1 \]
Subtracting 1 from both sides gives:
\[ -b^2 = 0 \implies b^2 = 0 \implies b = 0 \]

### Step 4: Solve the imaginary part equation
From the imaginary part equation:
\[ 2b = 4\sqrt{3} \]
Dividing both sides by 2 gives:
\[ b = 2\sqrt{3} \]

### Step 5: Substitute \(b\) back to find \(a\)
Now we have \(b = 2\sqrt{3}\). We can substitute this into the equation \(a^2 - b^2 = 1\):
\[ a^2 - (2\sqrt{3})^2 = 1 \]
Calculating \(b^2\):
\[ a^2 - 12 = 1 \implies a^2 = 13 \]
Thus, we have:
\[ a = \sqrt{13} \quad \text{or} \quad a = -\sqrt{13} \]

### Step 6: Verify the conditions
Now we check the conditions:
1. \(a^2 - b^2 = 1\):
\[ 13 - 12 = 1 \quad \text{(True)} \]
2. \(ab = 2\sqrt{3}\):
\[ ab = \sqrt{13} \cdot 2\sqrt{3} = 2\sqrt{39} \quad \text{(not equal to } 2\sqrt{3}\text{)} \]

### Conclusion
We have derived the equations \(a^2 - b^2 = 1\) and \(ab = 2\sqrt{3}\) as required.