A line segment is bisected by a line with the equation # 4 y + 3 x = 8 #. If one end of the line segment is at #( 1 , 8 )#, where is the other end?

1 Answer
May 16, 2016

Over the straight line #4y+3x+19=0#

Explanation:

The straight #4y+3x-8 = 3x +4(y-2)=0# passes by point #(0,2)# and has the direction given by the vector #v = (4,-3)#. In parametric form can be written as
#p = p_0+lambda v# with #p_0=(0,2)#
Given a generic straight point #p#, the symmetrical point to #q = (1,8)# regarding the straight line, is given by
#q_S = q +2(p-q) = 2p-q = 2p_0-q+2lambda v# which is a straight line parallel to the initial straight whose equation is given by
#q_S=2(0,2)-(1,8)+2\lambda(4,-3)#
In Cartesian coordinates we have
#x=-1+8\lambda#
#y = 4-8-6\lambda#
The Cartesian representation gives us
#4y+3x+19=0#