Question

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

• A. $2b$
• B. $2(a - b)$
• C. $\sqrt{2a^2 + 2b^2 - 4ab}$
• D. $\sqrt{2a^2 + 2b^2}$

Hint:

In coordinate geometry, always expand step-by-step using the distance formula to avoid sign mistakes, especially with negative coordinates.

Answer 1

The Correct Option is D

Step 1: Recall the distance formula.
The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 2: Substitute the given points.
Let \( (x_1, y_1) = (a, b) \) and \( (x_2, y_2) = (b, -a) \).
\[ \text{Distance} = \sqrt{(b - a)^2 + (-a - b)^2} \]
Step 3: Expand both squares.
\[ (b - a)^2 = b^2 - 2ab + a^2 \] \[ (-a - b)^2 = a^2 + 2ab + b^2 \]
Step 4: Add them.
\[ (b - a)^2 + (-a - b)^2 = (b^2 - 2ab + a^2) + (a^2 + 2ab + b^2) = 2a^2 + 2b^2 \] Step 5: Take square root.
\[ \text{Distance} = \sqrt{2a^2 + 2b^2} \] Step 6: Conclusion.
The required distance is \( \boxed{\sqrt{2a^2 + 2b^2}} \).