Question

Find the distance between the following pair of points:

(a sin α, –b cos α) and (–a cos α, b sin α)

Answer 1

The distance d between two points (x₁, y₁) and (x₂, y₂) is given by the formula.

`d = sqrt((x_1 - x_2)^2 + (y_1 - y_2)^2)`

The two given points are (a sin α, –b cos α) and (–a cos α, b sin α)

The distance between these two points is

`d = sqrt((a sin alpha + a cos alpha)^2 + (- b cos alpha - bsin alpha)^2)`

`= sqrt(a^2(sin alpha + cos alpha)^2 + b^2(-1)^2(cos alpha + sin alpha)^2)`

`= sqrt(a^2(sin alpha + cos alpha)^2 + b^2(sin alpha + cos alpha)^2)`

`= sqrt((a^2 + b^2)(sin alpha + cos alpha)^2)`

`d = (sin alpha + cos alpha)sqrt((a^2 + b^2))`

Hence the distance is `(sin alpha + cos alpha)sqrt((a^2 + b^2))`