# 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?

May 16, 2016

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

#### Explanation:

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