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

1 Answer
May 24, 2018

#color(blue)((-12/5,29/5)#

Explanation:

We know that the line #-y+3x=1# and the line containing the point #(6.3)# are perpendicular. If two lines are perpendicular then the product of their gradients is #-1#

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

This has a gradient of 3. The line containing #(6,3)# therefore has a gradient:

#m*3=-1=>m=-1/3#

Finding the equation of this line using point slope method:

#y-3=-1/3(x-6)=>y=-1/3x+5 \ \ \ \[2]#

Solving #[1] and [2]# simultaneously:

#-1/3x+5-3x+1=0=>x=9/5#

Substitute in #[1]#

#y=3(9/5)-1=22/5#

#(9/5,22/5)# are the coordinates of the midpoint.

Coordinates of the midpoint are found using:

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

Let the unknown endpoint be #(x_2,y_2)#

Then:

#((6+x_2)/2,(3+y_2)/2)->(9/5,22/5)#

#(6+x_2)/2=9/5=>x_2=-12/5#

#(3+y_2)/2=22/5=>y_2=29/5#

Coordinates of the other endpoint are:

#(-12/5,29/5)#