A line segment is bisected by a line with the equation # - 2 y + 3 x = 1 #. If one end of the line segment is at #(6 ,3 )#, where is the other end?

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Answer 1 · 2018-05-14T13:32:44

#(12/13,83/13)#

#-2y+3x=1=>y=3/2x-1/2 \ \ \[1]#

If the line segment was a line perpendicular to #[1]#. then its gradient would be the negative reciprocal of the gradient of #[1]#

#-2/3#

Forming the equation of a line for this:

#y-3=-2/3(x-6)#

#y=-2/3x+7\ \ \ \ [2]#

Finding the intersection of #[1] and [2]#

#-2/3x+7=3/2x-1/2#

#x=45/13#

Substituting in #[1]#

#y=3/2(45/13)-1/2=61/13#

The coordinates of the midpoint are given by:

#((x_1+x_2)/2,(y_1+y_2)/2)#

We have #(6,3)#
#:.#

#((6+x_2)/2,(3+y_2)/2)=>(45/13,61/13)#

Hence:

#(6+x)/2=45/13=>x=12/13#

#(3+y)/2=61/13=>y=83/13#

Coordinates of the other end are:

#(12/13,83/13)#