The distance between the complex numbers ` 2 +i and -3 + 5i ` is

Options

A. (a) ` sqrt(41)`
B. (b) ` sqrt(14)`
C. (c) ` sqrt(40)`
D. (d) ` 2`

Answer

To find the distance between the complex numbers \( z_1 = 2 + i \) and \( z_2 = -3 + 5i \), we can treat these complex numbers as points in the Cartesian plane. The real part represents the x-coordinate and the imaginary part represents the y-coordinate.

### Step-by-Step Solution:

1. **Identify the complex numbers**:
- Let \( z_1 = 2 + i \) which corresponds to the point \( (2, 1) \) in the Cartesian plane.
- Let \( z_2 = -3 + 5i \) which corresponds to the point \( (-3, 5) \) in the Cartesian plane.

2. **Use the distance formula**:
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

3. **Substitute the coordinates**:
Here, \( (x_1, y_1) = (2, 1) \) and \( (x_2, y_2) = (-3, 5) \).
\[ d = \sqrt{((-3) - 2)^2 + (5 - 1)^2} \]

4. **Calculate the differences**:
- Calculate \( x_2 - x_1 \):
\[ -3 - 2 = -5 \]
- Calculate \( y_2 - y_1 \):
\[ 5 - 1 = 4 \]

5. **Square the differences**:
\[ d = \sqrt{(-5)^2 + (4)^2} \]
\[ = \sqrt{25 + 16} \]

6. **Add the squares**:
\[ = \sqrt{41} \]

7. **Final result**:
Thus, the distance between the complex numbers \( 2 + i \) and \( -3 + 5i \) is:
\[ d = \sqrt{41} \]

### Conclusion:
The distance between the complex numbers \( 2 + i \) and \( -3 + 5i \) is \( \sqrt{41} \).