A line segment is bisected by line with the equation # 6 y - 2 x = 1 #. If one end of the line segment is at #(2 ,5 )#, where is the other end?

1 Answer
Jan 26, 2017

#(4 1/2, -2 1/2)#

Explanation:

#6y-2x=1#
#6y=2x+1#

#y=1/3x+1/6#
The slope for this equation is #1/3#, therefor the slope for line segment,#m =-3# where #m*m_1=-1#

The equation of line segment is #(y-y_1)=m(x-x_1)# where #x_1=2, y_1=5#

#(y-5)=-3(x-2)#
#y=-3x+6+5=-3x+11# #->a#

so, the intercept between 2 lines is
#1/3x+1/6=-3x+11#

#2/6x+1/6=-3x+11#

#2x+1=6(-3x+11)#
#2x=-18x+66-1#
#20x=65#
#x = 65/20 = 13/4 =3 1/4#

therefore,
#y=-3(13/4)+11#
#y=-39/4+44/4#
#y=5/4=1 1/4#

The line which intercept with both lines is a midpoint of the line segment. Therefore the other end line #(x,y)#

#(x+2)/2=13/4#, #(y+5)/2=5/4#

#x+2=13/2#, #y+5=5/2#

#x=13/2-2#, #y=5/2-5#
#x=9/2=4 1/2#, #y=-5/2=-2 1/2#