The vertices B and D of a parallelogram are `1-2i` and `4-2i` If the diagonals are at right angles and AC=2BD, the complex number representing A is

Options

A. `5/2`
B. `3i-3/2`
C. `3i-4`
D. `3i+4`

Answer

To solve the problem step by step, we will follow the given conditions and properties of the parallelogram and its diagonals.

### Step 1: Identify the given points
The vertices B and D of the parallelogram are given as:
- \( B = 1 - 2i \)
- \( D = 4 - 2i \)

### Step 2: Find the midpoint O of diagonal BD
The midpoint O of the diagonal BD can be calculated using the formula for the midpoint of two complex numbers:
\[ O = \frac{B + D}{2} = \frac{(1 - 2i) + (4 - 2i)}{2} = \frac{5 - 4i}{2} = \frac{5}{2} - 2i \]

### Step 3: Calculate the length of BD
To find the length of BD, we can use the distance formula:
\[ BD = |D - B| = |(4 - 2i) - (1 - 2i)| = |3| = 3 \]

### Step 4: Use the condition AC = 2BD
Given that \( AC = 2BD \), we can find the length of AC:
\[ AC = 2 \times 3 = 6 \]

### Step 5: Use the right angle condition of the diagonals
Since the diagonals are at right angles, we can express the relationship between the points A, B, C, and D. The vector \( OA \) can be expressed as a rotation of \( OD \) by 90 degrees. The vector \( OD \) is:
\[ OD = D - O = (4 - 2i) - \left(\frac{5}{2} - 2i\right) = \frac{3}{2} \]
Rotating this vector by 90 degrees (multiplying by \( i \)):
\[ OA = i \cdot OD = i \cdot \frac{3}{2} = \frac{3}{2}i \]

### Step 6: Find the position of point A
Using the midpoint O, we can express point A:
\[ A = O + OA = \left(\frac{5}{2} - 2i\right) + \left(\frac{3}{2}i\right) = \frac{5}{2} + \left(-2 + \frac{3}{2}\right)i = \frac{5}{2} - \frac{1}{2}i \]

### Step 7: Finalize the complex number for A
Thus, the complex number representing point A is:
\[ A = \frac{5}{2} - \frac{1}{2}i \]

### Summary
The complex number representing A is:
\[ A = \frac{5}{2} - \frac{1}{2}i \]