Find the distance between the following pairs of points :
(a,b), (-a,-b)

Answer

To find the distance between the points (a, b) and (-a, -b), we can use the distance formula. The distance \( D \) 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 the first point be \( P_1 = (a, b) \) and the second point be \( P_2 = (-a, -b) \).

2. **Assign Coordinates**:
Here, we can assign:
- \( x_1 = a \), \( y_1 = b \)
- \( x_2 = -a \), \( y_2 = -b \)

3. **Apply the Distance Formula**:
Substitute the coordinates into the distance formula:
\[ D = \sqrt{((-a) - a)^2 + ((-b) - b)^2} \]

4. **Simplify the Expression**:
Calculate \( (-a) - a \) and \( (-b) - b \):
\[ D = \sqrt{(-2a)^2 + (-2b)^2} \]

5. **Square the Terms**:
Now square the terms:
\[ D = \sqrt{(4a^2) + (4b^2)} \]

6. **Factor Out Common Terms**:
Factor out the common term:
\[ D = \sqrt{4(a^2 + b^2)} \]

7. **Simplify Further**:
Since \( \sqrt{4} = 2 \), we can simplify:
\[ D = 2\sqrt{a^2 + b^2} \]

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