Question

Point A is at `(-2 ,-8 )` and point B is at `(-5 ,3 )`. Point A is rotated `(3pi)/2` clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?

Answer 1

Let Initial polar coordinate of A ,`(r,theta)`
Given Initial Cartesian coordinate of A ,`(x_1=-2,y_1=-8)`
So we can write
`(x_1=-2=rcosthetaandy_1=-8=rsintheta)`
After `3pi/2` clockwise rotation the new coordinate of A becomes
`x_2=rcos(-3pi/2+theta)=rcos(3pi/2-theta)=-rsintheta=-(-8)=8`

`y_2=rsin(-3pi/2+theta)=-rsin(3pi/2-theta)=rcostheta=-2`

Initial distance of A from B(-5,3)
`d_1=sqrt(3^2+11^2)=sqrt130`
final distance between new position of A(8,-2) and B(-5,3)
`d_2=sqrt(13^2+5^2)=sqrt194`
So Difference=`sqrt194-sqrt130`

also consult the link

http://socratic.org/questions/point-a-is-at-1-4-and-point-b-is-at-9-2-point-a-is-rotated-3pi-2-clockwise-about#238064