check the picture below, notice the ratios, from AP to PB
[tex]\bf \textit{internal division of a line segment}
\\ \quad \\
A(-6,-5)\qquad B(4,0)\qquad
ratio1=2\qquad ratio2=3\qquad 2:3\\ \quad \\ \quad \\
\cfrac{A{ P }}{{ P }B}=\cfrac{ratio1}{ratio2}\implies \cfrac{A{ }}{{ }B}=\cfrac{ratio1}{ratio2}\implies
ratio2\cdot A=ratio1\cdot B\quad \\\\\\
3(-6,-5)=2(4,0)
\\ \quad \\
{{ P=\left(\cfrac{\textit{sum of "x" values}}{ratio1+ratio2}\quad ,\quad \cfrac{\textit{sum of "y" values}}{ratio1+ratio2}\right)}}\\ \quad \\
[/tex]
[tex]\bf
\qquad thus\qquad \\ \quad \\
P=\left(\cfrac{(3\cdot -6)+(2\cdot 4)}{2+3}\quad ,\quad \cfrac{(3\cdot -5)+(2\cdot 0)}{2+3}\right)
[/tex]
