If `a + ib = c + id`, then

Options

A. `a^(2) + c^(2) = 0`
B. `b^(2)+ c^(2) = 0`
C. `b^(2) + d^(2) = 0`
D. `a^(2) + b^(2) = c^(2) + d^(2)`

Explanation

To solve the equation \( a + ib = c + id \), we will follow these steps:

### Step 1: Understand the Equation
We start with the equation:
\[ a + ib = c + id \]
This means that the complex number on the left side is equal to the complex number on the right side.

### Step 2: Separate Real and Imaginary Parts
From the equation, we can separate the real and imaginary parts:
- Real part: \( a = c \)
- Imaginary part: \( b = d \)

### Step 3: Take the Modulus of Both Sides
Next, we take the modulus of both sides of the equation:
\[ |a + ib| = |c + id| \]
The modulus of a complex number \( z = x + iy \) is defined as:
\[ |z| = \sqrt{x^2 + y^2} \]
So we can express the moduli as:
\[ |a + ib| = \sqrt{a^2 + b^2} \]
\[ |c + id| = \sqrt{c^2 + d^2} \]

### Step 4: Set the Moduli Equal
Now we set the moduli equal to each other:
\[ \sqrt{a^2 + b^2} = \sqrt{c^2 + d^2} \]

### Step 5: Square Both Sides
To eliminate the square roots, we square both sides:
\[ a^2 + b^2 = c^2 + d^2 \]

### Conclusion
Thus, we have derived the equation:
\[ a^2 + b^2 = c^2 + d^2 \]
This is the final result.