A line segment is bisected by a line with the equation # - 6 y + 5 x = 4 #. If one end of the line segment is at #( 2 , 5 )#, where is the other end?

1 Answer
Dec 31, 2017

coordinates of the other end point #color (purple)(362/61, 17/61)#

Explanation:

Assumption : Bisecting line is a perpendicular bisector

Standard form of equation #y=mx +c#
Slope of perpendicular bisector m is given by
#-6y + 5x= 4#
#y= (5/6)x - (2/3)#
#m = (5/6)#

Slope of line segment is
#y - 5 = -(1/m)(x-2)#
#y - 5= (-6/5)(x - 2)#
#5y - 25 = -6x + 12#

#5y + 6x = 37 color (white)((aaaa)# Eqn (1)
#-6y + 5x = 4 color (white)((aaaa)# Eqn (2)

Solving Eqns (1) & (2),

#x= color (green)(242/61)#

#y= color (green)(161/61)#

Mid point #color(green)(242/61, 161/61)#

Let (x1,y1) the other end point.
#(2+x1)/2= 242/37#
#x1= (484/61) - 2 = color(red)( 362/61)#
#(5+y1)/2= 161/61#
#y1=(322/61) - 5 color(red)(17/61)#

Coordinates of other end point #color(red)(362/61, 17/61)#