Given:
The point (4, 2) divides the line joining the points (2, -4) and (8, 14).
To find: The ratio
​Explanation:
Let the ratio be,
[tex]m:n=k:1[/tex]Using the section formula,
[tex]P=\lparen\frac{mx_2+nx_{1,}}{m+n},\frac{my_2+ny_1}{m+n})[/tex]Here, we have
[tex]\begin{gathered} m=k,n=1 \\ x_1=2,y_1=-4 \\ x_2=8,y_2=14 \end{gathered}[/tex]On substitution we get,
[tex](4,2)=\lparen\frac{8k+2}{k+1},\frac{14k+4}{k+1})[/tex]Equating the coordinates we get,
[tex]\begin{gathered} 4=\frac{8k+2}{k+1} \\ 4\left(k+1\right)=8k+2 \\ 4k+4=8k+2 \\ 4k=2 \\ k=\frac{1}{2} \end{gathered}[/tex]Since,
[tex]\begin{gathered} k=\frac{1}{2} \\ \therefore m:n=\frac{1}{2}:1 \\ m:n=1:2 \end{gathered}[/tex]Hence, the ratio in which the point (4, 2) divides the line joining the points (2, -4) and (8, 14) is 1: 2.
Final answer: The ratio is,
[tex]1:2[/tex]