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

Oct 28, 2016

P=(56/25;-167/25)

Explanation:

Calculate the normal passing per A=(8;1)
$m = - \frac{3}{4}$
${m}_{\vdash} = \frac{4}{3}$
Normal equation
$y - 1 = \frac{4}{3} \left(x - 8\right)$
$4 x - 3 y = 29$
Calculate intersection I
$\left\{\begin{matrix}3 x + 4 y = 4 \\ 4 x - 3 y = 29\end{matrix}\right.$
Solving the system you get I=(128/25;-71/25)
Calculate $P$ so that $I$ is the medium point of $\overline{P A}$
$\left\{\begin{matrix}\frac{{P}_{x} + 8}{2} = \frac{128}{25} \\ \frac{{p}_{y} + 1}{2} = - \frac{71}{25}\end{matrix}\right.$
${P}_{x} = \frac{56}{25}$
${P}_{y} = - \frac{167}{25}$