Question

Find the distance between the points:

P(a sin α, a cos α) and Q(a cos α, – a sin α)

Answer 1

P(a sin α, a cos α) and Q(a cos α, - a sin α)
The given points are P(a sin α, a cos α) and Q(a cos α, - a sin α)
`Then (x_1 = a sin , y_1 = a cos α) and (x_2 = a cos a, y_2 = - a sin α)`

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

`= sqrt(( a cos α - a sin α)^2 + ( -a sin α - a cos α)^2)`

`=sqrt((a^2 cos^2  α  + a^2 sin^2  α  - 2a^2 cos  α xx sin  α )+(a^2 sin^2  α  + a^2 cos^2  α +2a^2 cos  α xx sin  α))`

`= sqrt(2a^2 cos^2  α +2a^2 sin^2  α)`

`= sqrt(2a^2 (cos^2  α + sin^2  α ))`

`= sqrt( 2a^2 (1))                 ("From the identity" cos^2 α  + sin^2 α=1 )`

`=sqrt(2a^2)`

`= sqrt(2a)`   units