Determine the ratio in which the straight line $x – y – 2 = 0$ divides the line segment joining $(3, -1)$ and $(8, 9)$.

Given:

The straight line $x – y – 2 = 0$ divides the line segment joining $(3, -1)$ and $(8, 9)$.

To do:

We have to find the ratio of division.

Solution:

Let the straight line $x – y – 2 = 0$ divides the line segment joining the points $(3, -1), (8, 9)$ in the ratio $m : n$ at point $(x_1,y_1)$.

Using the section formula, if a point $( x,\ y)$ divides the line joining the points $( x_1,\ y_1)$ and $( x_2,\ y_2)$ in the ratio $m:n$, then 

$(x,\ y)=( \frac{mx_2+nx_1}{m+n},\ \frac{my_2+ny_1}{m+n})$

\( (x_1,y_1)=(\frac{m \times 8+n \times 3}{m+n}, \frac{m \times 9+n \times (-1)}{m+n}) \)

\( =(\frac{8 m+3 n}{m+n}, \frac{9 m-n}{m+n}) \)

The point \( (x, y) \) lies on the line \( x-y-2=0 \).

This implies, the point \( (x,y) \) satisfies the above equation.

\( \Rightarrow \frac{8 m+3 n}{m+n}-\frac{9 m-n}{m+n}-2=0 \)

\( \Rightarrow(8 m+3 n)-(9 m-n)-2(m+n)=0 \)

\( \Rightarrow 8 m+3 n-9 m+n-2 m-2 n=0 \)

\( \Rightarrow-3 m+2 n=0 \)

\( \Rightarrow 2 n=3 m \)

\( \Rightarrow \frac{m}{n}=\frac{2}{3} \)

\( \Rightarrow m:n=2:3 \)

The required ratio of division is $2:3$.

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Updated on: 10-Oct-2022

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