Question

Point A is at `(1 ,-4 )` and point B is at `(-9 ,-2 )`. 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

hew coordinate (4,1)
Change in distance `Delta=d_2-d_1=sqrt178-sqrt104`

Explanation:

Distance of initial position of the point A i.e. P (1.-4) from origin O is the radius of the circular path `r` = `sqrt((1-0)^2+(-4-0)^2)=sqrt17` not required

If the initial `/_XOP=-theta`
Initial x-coordinate of A at P is `x_1=1=rcos(-theta)=>rcos(theta)=1`
Initial y-coordinate of A at P is `y_1==-4=rsin(-theta)=>rsin(theta)=4`

After clockwise rotation at an angle `3pi/2 =270^o` in a circular path of radius ``the point takes new position Q having coordinate `(x_2,y_2)`
`x_2=rcos(-270-theta)=rcos(270+theta)=rsintheta=4`
`y_2=rsin(-270-theta)=-rsin(270+theta)=rcostheta=1`

The distance between P,Q
`d=sqrt((1-4)^2+(-4-1)^2)=sqrt34` not wanted
Initial distance between A(1,-4)i.e.P andB(-9-2)
`d_1=sqrt((1+9)^2+(-4+2)^2)=sqrt104`

Fnitial distance between A(4,1)i.e.Q andB(-9-2)
`d_2=sqrt((4+9)^2+(1+2)^2)=sqrt178`

Change in distance `Delta=d_2-d_1=sqrt178-sqrt104`