Find the ratio in which the line 3x+4y-9=0 divides the line segment joining the points(1?-3) and (2?7)

Dear student
Assuming the question as:
Find the ratio in which the line 3x+4y-9=0 divides the line segment joining the points(1,-3) and (2,7)
Solution:
$$\begin{gathered} \mathrm{Let} \mathrm{the} \mathrm{line} 3\mathrm{x}+4\mathrm{y}–9=0 \mathrm{divides} \mathrm{the} \mathrm{line} \mathrm{segment} \mathrm{joining} \mathrm{the} \mathrm{points} (1, -3) \mathrm{and} (2, 7) \mathrm{in} \mathrm{k}:1 \\ \mathrm{Now}, \mathrm{the} \mathrm{coordinate} \mathrm{of} \mathrm{the} \mathrm{point} \mathrm{that} \mathrm{divides} \mathrm{the} \mathrm{line} \mathrm{joining} \mathrm{the} \mathrm{points} (1,- 3) \mathrm{and} (2, 7) \mathrm{in} \mathrm{k}:1 \mathrm{are} \\ \left(\frac{\mathrm{k}\left(2\right)+1\left(1\right)}{\mathrm{k}+1},\frac{\mathrm{k}\left(7\right)+1\left(-3\right)}{\mathrm{k}+1}\right) \left[\mathrm{using} \mathrm{section} \mathrm{formula}\right] \\ =\left(\frac{2\mathrm{k}+1}{\mathrm{k}+1},\frac{7\mathrm{k}-3}{\mathrm{k}+1}\right) \\ \mathrm{The} \mathrm{point} \left(\frac{2\mathrm{k}+1}{\mathrm{k}+1},\frac{7\mathrm{k}-3}{\mathrm{k}+1}\right) \mathrm{is} \mathrm{the} \mathrm{point} \mathrm{of} \mathrm{intersection} \mathrm{of} \mathrm{the} \mathrm{line} 3\mathrm{x}+4\mathrm{y}-9=0 \\ \mathrm{and} \mathrm{the} \mathrm{line} \mathrm{segment} \mathrm{joining} \mathrm{the} \mathrm{points} (1, -3) \mathrm{and} (2, 7) \\ \mathrm{Thus}, \mathrm{the} \mathrm{point} \left(\frac{2\mathrm{k}+1}{\mathrm{k}+1},\frac{7\mathrm{k}-3}{\mathrm{k}+1}\right) \mathrm{lies} \mathrm{on} \mathrm{the} \mathrm{line} 3\mathrm{x}+4\mathrm{y}-9=0 \\ \mathrm{So}, \mathrm{we} \mathrm{have} \\ 3\left(\frac{2\mathrm{k}+1}{\mathrm{k}+1}\right)+4\left(\frac{7\mathrm{k}-3}{\mathrm{k}+1}\right)-9=0 \\ \Rightarrow 3\left(2\mathrm{k}+1\right)+4\left(7\mathrm{k}-3\right)-9\left(\mathrm{k}+1\right)=0 \\ \Rightarrow 6\mathrm{k}+3+28\mathrm{k}-12-9\mathrm{k}-9=0 \\ \Rightarrow 25\mathrm{k}-18=0 \\ \Rightarrow \mathrm{k}=\frac{18}{25} \\ \mathrm{This} \mathrm{means} \mathrm{the} \mathrm{line} 3\mathrm{x}+4\mathrm{y}–9=0 \mathrm{divides} \mathrm{the} \mathrm{line} \mathrm{segment} \mathrm{joining} \mathrm{the} \mathrm{points} (1, -3) \mathrm{and} (2, 7) \mathrm{in} \mathrm{the} \mathrm{ratio} 18: 25 \end{gathered}$$
​​​​​Regards