If `P, Q, R, S` are represented by the complex number `4 +i,1+6i,-4+3i,-1-2i` respectively, then `PQRS` is a (A) rectangle (B) square (C) rhombus (D) parallelogram

Answer

To determine the shape formed by the points \( P, Q, R, S \) represented by the complex numbers \( 4 + i, 1 + 6i, -4 + 3i, -1 - 2i \) respectively, we will follow these steps:

### Step 1: Identify the coordinates of the points
The complex numbers can be converted to Cartesian coordinates:
- \( P(4, 1) \)
- \( Q(1, 6) \)
- \( R(-4, 3) \)
- \( S(-1, -2) \)

### Step 2: Calculate the lengths of the sides
We will calculate the distances between consecutive points \( PQ, QR, RS, SP \).

#### Distance \( PQ \):
\[ PQ = \sqrt{(4 - 1)^2 + (1 - 6)^2} = \sqrt{3^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \]

#### Distance \( QR \):
\[ QR = \sqrt{(1 - (-4))^2 + (6 - 3)^2} = \sqrt{(1 + 4)^2 + (6 - 3)^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \]

#### Distance \( RS \):
\[ RS = \sqrt{(-4 - (-1))^2 + (3 - (-2))^2} = \sqrt{(-4 + 1)^2 + (3 + 2)^2} = \sqrt{(-3)^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \]

#### Distance \( SP \):
\[ SP = \sqrt{(-1 - 4)^2 + (-2 - 1)^2} = \sqrt{(-5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34} \]

### Step 3: Check the lengths of the sides
We find that:
\[ PQ = QR = RS = SP = \sqrt{34} \]
Since all four sides are equal, the figure could be a rhombus or a square.

### Step 4: Calculate the lengths of the diagonals
Next, we will calculate the lengths of the diagonals \( PR \) and \( QS \).

#### Diagonal \( PR \):
\[ PR = \sqrt{(4 - (-4))^2 + (1 - 3)^2} = \sqrt{(4 + 4)^2 + (1 - 3)^2} = \sqrt{8^2 + (-2)^2} = \sqrt{64 + 4} = \sqrt{68} \]

#### Diagonal \( QS \):
\[ QS = \sqrt{(1 - (-1))^2 + (6 - (-2))^2} = \sqrt{(1 + 1)^2 + (6 + 2)^2} = \sqrt{2^2 + 8^2} = \sqrt{4 + 64} = \sqrt{68} \]

### Step 5: Check the lengths of the diagonals
We find that:
\[ PR = QS = \sqrt{68} \]
Since both diagonals are equal and all sides are equal, the figure is a square.

### Conclusion
The shape formed by the points \( P, Q, R, S \) is a square.

### Answer
(B) square