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?

1 Answer
P dilip_k
Mar 11, 2016

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

Explanation:

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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

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