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

Hint: 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 \[\left( {\dfrac{{1.{x_2} + k.{x_1}}}{{1 + k}},\dfrac{{1.{y_2} + k.{y_1}}}{{1 + k}}} \right)\].