Question

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?

Answer 1

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`