The distance between the points (a, b) and (-a, -b)

Options

A. 0
B. 1
C. `sqrt(ab)`
D. `2sqrt(a^(2) + b^(2))`

Answer

To find the distance between the points (a, b) and (-a, -b), we can use the distance formula in coordinate geometry. The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step-by-step Solution:

1. **Identify the Points**:
- Let point P be \( (a, b) \) and point Q be \( (-a, -b) \).

2. **Substitute the Coordinates into the Distance Formula**:
- Here, \( x_1 = a \), \( y_1 = b \), \( x_2 = -a \), and \( y_2 = -b \).
- Substitute these values into the distance formula:
\[ d = \sqrt{((-a) - a)^2 + ((-b) - b)^2} \]

3. **Simplify the Expressions**:
- Calculate \( (-a) - a = -2a \) and \( (-b) - b = -2b \).
- Now, substitute these back into the formula:
\[ d = \sqrt{(-2a)^2 + (-2b)^2} \]

4. **Calculate the Squares**:
- \( (-2a)^2 = 4a^2 \) and \( (-2b)^2 = 4b^2 \).
- Therefore, we have:
\[ d = \sqrt{4a^2 + 4b^2} \]

5. **Factor Out the Common Terms**:
- Factor out 4 from the square root:
\[ d = \sqrt{4(a^2 + b^2)} \]

6. **Simplify the Square Root**:
- Since \( \sqrt{4} = 2 \), we can simplify further:
\[ d = 2\sqrt{a^2 + b^2} \]

### Final Answer:
The distance between the points \( (a, b) \) and \( (-a, -b) \) is \( 2\sqrt{a^2 + b^2} \).

---