Question

Find the co-ordinates of the point which divides the line segment joining the points $A(3,4)$ and $B(-2,-1)$ in the ratio $3:2$.

Hint:

When dividing a line in ratio $m:n$, always apply the section formula: $\Big(\tfrac{mx_2+nx_1}{m+n},\tfrac{my_2+ny_1}{m+n}\Big)$.

Answer 1

Let $P(x,y)$ be the required point. If $P$ divides $AB$ in the ratio $m:n=3:2$, then by section formula: \[ x = \frac{mx_2 + nx_1}{m+n}, y = \frac{my_2 + ny_1}{m+n} \] Here $x_1=3, \ y_1=4, \ x_2=-2, \ y_2=-1, \ m=3, \ n=2$. \[ x = \frac{3(-2)+2(3)}{3+2} = \frac{-6+6}{5}=0 \] \[ y = \frac{3(-1)+2(4)}{3+2} = \frac{-3+8}{5}=\frac{5}{5}=1 \] \[ \boxed{P(0,1)} \]