Question

In what ratio the line $y-x+2=0$ divides the line joining the points $(3,-1)$ and $(8,9)$

• A. $1:2$
• B. $2:1$
• C. $2:3$
• D. $3:4$

Answer 1

The correct option is C

$2:3$

Explanation for the correct option:

Finding the ratio of the line:

Consider the points $A(3,-1)=(x_2,y_2)$and $B(8,9)=(x_1,y_1)$

Consider the line segments joining the points $A$ and $B$ divided in ratio $m:1$at point $C$.
By section formula,

$x=\frac{m{x}_1+n{x}_2}{m+n}$, and $y=\frac{m{y}_1+n{y}_2}{m+n}$

Then coordinates of $C=\left(\frac{8m+3}{m+1},\frac{9m-1}{m+1}\right)\left[\because n=1\right]$

As point C lies on the line $y-x+2=0$

$$\begin{gathered} \therefore \frac{9m-1}{m+1}-\frac{8m+3}{m+1}+2=0 \\ \Rightarrow 9m-1-8m-3+2m+2=0 \\ \Rightarrow 3m-2=0 \\ \Rightarrow m=\frac{2}{3} \end{gathered}$$
Thus the required ratio is $2:3$

Hence, the correct answer is option (C).