A line segment is bisected by a line with the equation # - 6 y + x = 3 #. If one end of the line segment is at #( 4 , 5 )#, where is the other end?

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Answer 1 · 2017-10-18T20:56:58

Midpoint #x = 9/37, y = -17/37#
Other end point #(-130/37, -219/37)#

Assumption : line segment is bisected by the line with equation #-6y+x = 3# at right angle.

#x - 6y = 3 color (white)(aaa) Eqn (1)#

#y=(1/6)(x - 3)# slope of line is #1/6#

Slope of perpendicular line is #-1/(1/6) = -6#

Equation of line segment is #(y - 5) = -6(x - 4)#

#6x + y = 1 color (white)(aaa)# Eqn (2)

Solving Eqns (1) & (2) we get the midpoint.
#37x = 9, x = 9/37#
#y = -17/37#

Let the other end point be #(x_1, y_1)#

#(x_1+4)/2 = 9/37; x_1= —130/37#
#(y_1 + 5)/2 = -17/37; y_1 = -219/37#