A line segment is bisected by a line with the equation # - 3 y + 4 x = 6 #. If one end of the line segment is at #( 3 , 1 )#, where is the other end?

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Answer 1 · 2017-06-21T09:58:48

The other end is #=(51/25,129/75)#

Let the other end be #(x,y)#

Then the mid-point of the line is #=((x+3)/2, (y+1)/2)#

The slope of the line #-3y+4x=6# ............#(1)#

is

#=4/3#

The slope of the line perpendicular is

#=-3/4#

The equation of the line is

#-3/4(x-3)=(y-1)#

#-3x+9=4y-4#

#4y+3x=13# .............#(2)#

Solving for #x# and #y#, in equations #(1)# and #(2)#

#4x-3*(13-3x)/4=6#

#16x-39+9x=24#

#25x=24+39=63#

#x=63/25#

#y=1/3*(4*63/25-6)#

#=102/75#

The mid-point is #=(63/25,102/75)#

So,

#(x+3)/2=63/25#

#x=126/25-3=51/25#

#(y+1)/2=102/75#

#y=204/75-1=129/75#

The other end is #=(51/25,129/75)#