Answer: The required co-ordinates of point C are (9, -3.5).
Step-by-step explanation: We are given the points A(3, -5) and B(19, -1).
We are to find the co-ordinates of point C that sit 3/8 of the way along AB, where the point P is close to A than to B.
According to the given information, we have
[tex]\dfrac{AC}{AB}=\dfrac{3}{8}\\\\\Rightarrow \dfrac{AC}{AC+BC}=\dfrac{3}{8}\\\\\Rightarrow 8AC=3AC+3BC\\\\\Rightarrow 5AC=3BC\\\\\Rightarrow AC:BC=3:5.[/tex]
So, point C divides the line segment AB in the ratio 3 : 5.
We know that
if a point Q divides a line segment joining the points S(a,b) and T(c,d), in the ratio m : n, then the co-ordinates of Q are
[tex]\left(\dfrac{mc+na}{m+n},\dfrac{md+nb}{m+n}\right).[/tex]
Therefore, the co-ordinates of point C are
[tex]\left(\dfrac{3\times19+5\times3}{3+5},\dfrac{3\times(-1)+5\times(-5)}{3+5}\right)\\\\\\=\left(\dfrac{57+15}{8},\dfrac{-3-25}{8}\right)\\\\=(9,-3.5).[/tex]
Thus, the required co-ordinates of point C are (9, -3.5).
