Question

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

Answer 1

Given: Points P(a, b) and Q(–b, a). Use the distance formula `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`.

Step-wise calculation:

1. x₂ – x₁ = (–b) – a

= –(a + b)

y₂ – y₁ = a – b

2. Distance = `sqrt(-(a + b)^2 + (a - b)^2)`

= `sqrt((a + b)^2 + (a - b)^2`

= `sqrt((a^2 + 2ab + b^2) + (a^2 - 2ab+ b^2)`

= `sqrt(2a^2 + 2b^2)`

= `sqrt(2(a^2 + b^2)`

The distance between (a, b) and (–b, a) is `sqrt(2(a^2 + b^2)` equivalently `sqrt(2) · sqrt(a^2 + b^2)`.