Question

The ratio in which the line $x + y = 4$ divides the line joining the points $\left( { - 1,1} \right)$ and $\left( {5,7} \right)$ is
A.$1:2$
B.$2:1$
C.$3:2$
D.$3:1$

Answer 1

The equation of a straight line passing through two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ is $\dfrac{{y - {y_1}}}{{{y_2} - {y_1}}} = \dfrac{{x - {x_1}}}{{{x_2} - {x_1}}}$.
If the point $\left( {x,y} \right)$ divides the line segment joining the points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ in ratio $1:k$, then $x = \dfrac{{1.{x_2} + k.{x_1}}}{{1 + k}}$ and $y = \dfrac{{1.{y_2} + k.{y_1}}}{{1 + k}}$.

Complete step-by-step answer:
Here, the given points are $\left( { - 1,1} \right)$ and $\left( {5,7} \right)$. Therefore, the equation of the straight line joining the points $\left( { - 1,1} \right)$ and $\left( {5,7} \right)$ is

$$\dfrac{{y - 1}}{{7 - 1}} = \dfrac{{x - ( - 1)}}{{5 - ( - 1)}} \\\ \Rightarrow \dfrac{{y - 1}}{6} = \dfrac{{x - ( - 1)}}{6} \\\ \Rightarrow y - 1 = x + 1 \\\ \Rightarrow x - y + 2 = 0...............(1) \\\$$

The equation of the given line is
$x + y - 4 = 0...............(2)$
Now, adding Eq. (1) and Eq. (2) we get,

$$2x - 2 = 0 \\\ \Rightarrow x = 1 \\\$$

Therefore, from Eq. (2) we get, $y = (4 - x) = (4 - 1) = 3$
So, the point of intersection of that two lines is $\left( {1,3} \right)$.
Now, let the point $\left( {1,3} \right)$ divide the line segment joining $\left( { - 1,1} \right)$ and $\left( {5,7} \right)$ in ratio $1:k$.
Therefore, by the section formula, if $\left( {x,y} \right)$ divides the line segment joining the points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ in ratio $1:k$, then

$$x = \dfrac{{1.{x_2} + k.{x_1}}}{{1 + k}} \\\ \Rightarrow 1 = \dfrac{{1 \times 5 + k \times ( - 1)}}{{1 + k}} \\\ \Rightarrow 1 + k = 5 - k \\\ \Rightarrow 2k = 4 \\\ \Rightarrow k = 2 \\\$$ $$y = \dfrac{{1.{y_2} + k.{y_1}}}{{1 + k}} \\\ \Rightarrow 3 = \dfrac{{1 \times 7 + k \times 1}}{{1 + k}} \\\ \Rightarrow 3k + 3 = 7 + k \\\ \Rightarrow 2k = 4 \\\ \Rightarrow k = 2 \\\$$

Hence, the line joining the points $\left( { - 1,1} \right)$ and $\left( {5,7} \right)$ is divided by the line $x + y = 4$ in the ratio of $1:2$.

Note: You have to know the formula properly which are
The equation of a straight line passing through two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ is $\dfrac{{y - {y_1}}}{{{y_2} - {y_1}}} = \dfrac{{x - {x_1}}}{{{x_2} - {x_1}}}$. We can also express this equation in a different way as $\dfrac{{y - {y_1}}}{{x - {x_1}}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$.
The co-ordinate of a point which divides the straight line joining the two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ in ratio $1:k$ is $\left( {\dfrac{{1.{x_2} + k.{x_1}}}{{1 + k}},\dfrac{{1.{y_2} + k.{y_1}}}{{1 + k}}} \right)$.