Find distance between the points A (a,b) and B( - b,a)

Answer

To find the distance between the points A(a, b) and B(-b, a), we will use the distance formula. The distance formula between two points (x1, y1) and (x2, y2) is given by:

\[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]

### Step-by-Step Solution:

1. **Identify the coordinates:**
- Point A has coordinates (a, b), so \(x_1 = a\) and \(y_1 = b\).
- Point B has coordinates (-b, a), so \(x_2 = -b\) and \(y_2 = a\).

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

3. **Simplify the expressions inside the square root:**
- For the first term:
\[ (-b - a)^2 = (-(b + a))^2 = (b + a)^2 \]
- For the second term:
\[ (a - b)^2 \]

4. **Combine the two squared terms:**
\[ d = \sqrt{(b + a)^2 + (a - b)^2} \]

5. **Expand both squared terms:**
- Using the formula \((x + y)^2 = x^2 + 2xy + y^2\):
\[ (b + a)^2 = b^2 + 2ab + a^2 \]
- Using the formula \((x - y)^2 = x^2 - 2xy + y^2\):
\[ (a - b)^2 = a^2 - 2ab + b^2 \]

6. **Combine the expanded terms:**
\[ d = \sqrt{(b^2 + 2ab + a^2) + (a^2 - 2ab + b^2)} \]
\[ = \sqrt{2a^2 + 2b^2} \]

7. **Factor out the common term:**
\[ d = \sqrt{2(a^2 + b^2)} \]

8. **Final expression for the distance:**
\[ d = \sqrt{2} \cdot \sqrt{a^2 + b^2} \]

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