Question

Find the distance between the points \( (a, b) \) and \( (-a, -b) \).

Hint:

Use the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) to find the distance between two points.

Answer 1

To find the distance between the points \( (a, b) \) and \( (-a, -b) \), we can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}, \] where \( (x_1, y_1) = (a, b) \) and \( (x_2, y_2) = (-a, -b) \). Substitute the coordinates into the formula: \[ d = \sqrt{((-a) - a)^2 + ((-b) - b)^2} = \sqrt{(-2a)^2 + (-2b)^2}. \] Simplifying: \[ d = \sqrt{4a^2 + 4b^2} = \sqrt{4(a^2 + b^2)} = 2\sqrt{a^2 + b^2}. \]
Conclusion: The distance between the points \( (a, b) \) and \( (-a, -b) \) is \( 2\sqrt{a^2 + b^2} \).