Wyznaczmy równanie prostej AB.

$a_{AB}=\frac{y_b-y_a}{x_b-x_a}=\frac{14-\left(-6\right)}{8-\left(-4\right)}=\frac{20}{12}=\frac{10}{6}=\frac{5}{3}$  

Zatem prosta jest postaci:    

$y=\frac{5}{3}x+b$  

Podstawiając współrzędne punktu A mamy:

$-6=\frac{5}{3}\cdot \left(-4\right)+b$  

$-6=-\frac{20}{3}+b$  

$-\frac{18}{3}+\frac{20}{3}=b$  

$\frac{2}{3}=b$  

Zatem mamy:

$y=\frac{5}{3}x+\frac{2}{3}$  

Wobec tego punkt P jest postaci:

$P\left(x,\frac{5}{3}x+\frac{2}{3}\right)$  

Zał:

$-4<x<8$  

 

$\frac{\left|AP\right|}{\left|PB\right|}=\frac{3}{5}$  

$5\left|AP\right|=3\left|PB\right|$  

$5\sqrt{{\left(x-\left(-4\right)\right)}^2+{\left(\frac{5}{3}x+\frac{2}{3}-\left(-6\right)\right)}^2}=3\sqrt{{\left(8-x\right)}^2+{\left(14-\left(\frac{5}{3}x+\frac{2}{3}\right)\right)}^2}{\mid }^2$  

$25\left({\left(x+4\right)}^2+{\left(\frac{5}{3}x+\frac{2}{3}+\frac{18}{3}\right)}^2\right)=9\left({\left(8-x\right)}^2+{\left(14-\frac{5}{3}x-\frac{2}{3}\right)}^2\right)$  

$25\left(x^2+8x+16+{\left(\frac{5}{3}x+\frac{20}{3}\right)}^2\right)=9\left(64-16x+x^2+{\left(\frac{40}{3}-\frac{5}{3}x\right)}^2\right)$  

$25\left(x^2+8x+16+\frac{25}{9}{x}^2+\frac{200}{9}x+\frac{400}{9}\right)=9\left(64-16x+x^2+\frac{1600}{9}-\frac{400}{9}x+\frac{25}{9}{x}^2\right)$  

$\cancel{25x^2}+200x+400+\frac{625}{9}{x}^2+\frac{5000}{9}x+\frac{10000}{9}=576-144x+9x^2+1600-400x\cancel{+25x^2}$  

$200x+400+\frac{625}{9}{x}^2+\frac{5000}{9}x+\frac{10000}{9}=576-144x+9x^2+1600-400x\mid -200x-400$  

$\frac{625}{9}{x}^2+\frac{5000}{9}x+\frac{10000}{9}=1776-744x+9x^2\mid \cdot 9$  

$625x^2+5000x+10000=15984-6696x+81x^2$  

$544x^2+11696x-5984=0\mid :16$  

$34x^2+731x-374=0$  

$\Delta ={731}^2-4\cdot 34\cdot \left(-374\right)=534361+50864=585225$  

$\sqrt{\Delta }=\sqrt{585225}=765$  

$x_1=\frac{-731-765}{2\cdot 34}=\frac{-1496}{68}=-22\notin \left(-4,8\right)$  

$x_2=\frac{-731+765}{2\cdot 34}=\frac{34}{68}=\frac{1}{2}$  

Zatem mamy:

$\{\begin{array}{l}x=\frac{1}{2}\\ y=\frac{5}{3}x+\frac{2}{3}\end{array}$  

$\{\begin{array}{l}x=\frac{1}{2}\\ y=\frac{5}{3}\cdot \frac{1}{2}+\frac{2}{3}\end{array}$  

$\{\begin{array}{l}x=\frac{1}{2}\\ y=\frac{5}{6}+\frac{4}{6}\end{array}$  

$\{\begin{array}{l}x=\frac{1}{2}\\ y=\frac{9}{6}\end{array}$  

$\{\begin{array}{l}x=\frac{1}{2}\\ y=\frac{3}{2}\end{array}$  

$P\left(\frac{1}{2},\frac{3}{2}\right)$