A line segment is bisected by line with the equation # 6 y - 7 x = 3 #. If one end of the line segment is at #(4 ,2 )#, where is the other end?

1 Answer
May 16, 2018

#color(blue)((74/85,398/85)#

Explanation:

First we note that if two lines are perpendicular then the product of their gradients is #-1#

We need to find the equation of a line that contains the point #(4,2)# and is perpendicular to #6y-7x=3#

Rearranging # \ \ \ \6y-7x=3#

#y=7/6x+1/2 \ \ \ \[1]#

If the gradient for our given line be #m#, then:

#m*7/6=-1=>m=-6/7#

Using point slope form of a line:

#y-2=-6/7(x-4)#

#y=-6/7x+38/7 \ \ \ \[2]#

Finding the point of intersection.

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

#-6/7x+38/7-7/6x-1/2=0#

#x=207/85#

Substituting in #[1]#

#7/6(207/85)+1/2=284/85#

#(207/85,284/85)# are the coordinates of the midpoint:

The coordinates for the midpoint of a line is given by:

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

Therefore:

#((4+x_2)/2,(2+y_2)/2)->(207/85,284/85)#

Hence:

#(4+x)/2=207/85=>x=74/85#

#(2+y)/2=284/85=>y=398/85#

Coordinates of the other end are:

#(74/85,398/85)#

PLOT:

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