POQ is a straight line through the origin O,P and Q represent the complex numbers a+ib and c+id respectively and OP=OQ. Then, which one of the following is true?

Options

A. `|a+ib|=|c+id|`
B. `a+b=c+d`
C. `"arg"(a+ib)="arg"(c+id)`
D. none of these

Answer

To solve the problem, we need to analyze the given conditions about the complex numbers representing points P and Q on the Argand plane.

### Step-by-Step Solution:

1. **Understanding the Points**:
- Let \( P \) be represented by the complex number \( z_1 = a + ib \).
- Let \( Q \) be represented by the complex number \( z_2 = c + id \).
- Both points lie on a straight line through the origin \( O \).

2. **Distance from Origin**:
- The distance from the origin \( O \) to point \( P \) is given by the modulus of \( z_1 \), which is \( |z_1| = |a + ib| \).
- The distance from the origin \( O \) to point \( Q \) is given by the modulus of \( z_2 \), which is \( |z_2| = |c + id| \).

3. **Equality of Distances**:
- We are given that \( OP = OQ \). This means that the distances from the origin to points \( P \) and \( Q \) are equal:
\[ |z_1| = |z_2| \]
- Therefore, we have:
\[ |a + ib| = |c + id| \]

4. **Magnitude Calculation**:
- The magnitude of a complex number \( x + iy \) is calculated as:
\[ |x + iy| = \sqrt{x^2 + y^2} \]
- Applying this to our complex numbers:
\[ |a + ib| = \sqrt{a^2 + b^2} \]
\[ |c + id| = \sqrt{c^2 + d^2} \]

5. **Setting the Magnitudes Equal**:
- From the equality of distances, we can set up the equation:
\[ \sqrt{a^2 + b^2} = \sqrt{c^2 + d^2} \]

6. **Squaring Both Sides**:
- To eliminate the square roots, we square both sides:
\[ a^2 + b^2 = c^2 + d^2 \]

7. **Conclusion**:
- The condition that must hold true given the problem statement is:
\[ a^2 + b^2 = c^2 + d^2 \]
- This means that the sum of the squares of the real and imaginary parts of \( z_1 \) is equal to the sum of the squares of the real and imaginary parts of \( z_2 \).

### Final Answer:
The correct condition is \( a^2 + b^2 = c^2 + d^2 \).

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