Point A is at #(4 ,-2 )# and point B is at #(1 ,-3 )#. Point A is rotated #pi/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?

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Answer 1 · 2016-02-17T03:05:28

The new point #A(-2, -4)#
and No change in distance. Same distance#=sqrt10#

the equation passing thru the origin (0, 0) and point A(4, -2) is

#y=-1/2 x#

and distance from (0, 0) to A(4, -2) is #=2sqrt5#

the equation passing thru the origin (0, 0) and perpendicular to line #y=-1/2 x# is

#y=2x" " "#first equation

We are looking for a new point #A(x_1, y_1)# on the line #y=2x# at the 3rd quadrant and #2sqrt5# away from the origin (0, 0).

Use distance formula for "distance from line to a point not on the line"

#d=(ax_1+by_1+c)/(+-sqrt(a^2+b^2)#

Use equation, #y=-1/2x# which is also #x+2y=0#

#a=1# and #b=2# and #c=0#

#d=2sqrt5=(x_1+2y_1)/(+-sqrt(1^2+2^2))#

#2sqrt5=(x_1+2y_1)/(-sqrt(5))" " "#choose negative because,
#(x_1, y_1)# is below the line.

#-2sqrt5*sqrt5=x_1+2y_1#

#-10=x_1+2y_1" " "#second equation

#y_1=2x_1" " "#first equation

Using first and second equation. Solve for #(x_1, y_1)#

and #x_1=-2# and #y_1=-4#