Determine the ratio in which line $ 2x + y - 4 = 0 $ divides the line segment joining the points $ A\left( {2, - 2} \right)\,\,and\,\,B\left( {3,7} \right) $

Hint: For this we first let the ratio be $ k:1 $ then using section formula we first find coordinates of the point which is common on a given line segment and on a given line. Then substituting these coordinates in the given equation of a line to find the value of k or required ratio.
Formulas used: Section formula: $ x = \dfrac{{m{x_2} + n{x_1}}}{{m + n}},\,\,y = \dfrac{{m{y_2} + n{y_1}}}{{m + n}} $

Complete step-by-step answer:
Let the ratio be $ k:1 $ in which given line $ 2x + y - 4 = 0 $ divides the line segment joining the points
 $ A\left( {2, - 2} \right)\,\,and\,\,B\left( {3,7} \right) $ .

Then, by using the section formula we can find the coordinate of the point which divides the line segment joining the points $ A\left( {2, - 2} \right)\,\,and\,\,B\left( {3,7} \right) $ .
Therefore, we have:
 $ x = \dfrac{{3k + 2}}{{k + 1}},\,\,y = \dfrac{{7k - 2}}{{k + 1}} $
Since, this point is common on line segment joining points $ A\left( {2, - 2} \right)\,\,and\,\,B\left( {3,7} \right) $ and line $ 2x + y - 4 = 0 $ .
Therefore, substituting the value of ‘x’ and ‘y’ in the given equation and simplifying to get the value of k.
 $
  2\left( {\dfrac{{3k + 2}}{{k + 1}}} \right) + \left( {\dfrac{{7k - 2}}{{k + 1}}} \right) - 4 = 0 \\
   \Rightarrow \dfrac{{6k + 4}}{{k + 1}} + \dfrac{{7k - 2}}{{k + 1}} - 4 = 0 \\
  $
 $
  taking\,\,LCM \\
  \dfrac{{6k + 4 + 7k - 2 - 4\left( {k + 1} \right)}}{{k + 1}} = 0 \\
  $

 $
   \Rightarrow \dfrac{{13k + 2 - 4k - 4}}{{k + 1}} = 0 \\
   \Rightarrow \dfrac{{9k - 2}}{{k + 1}} = 0 \\
   \Rightarrow 9k - 2 = 0 \\
   \Rightarrow 9k = 2 \\
   \Rightarrow k = \dfrac{2}{9} \;
  $
Therefore, from above we can see that the value of k is $ \dfrac{2}{9} $ or we can say that ratio is $ \dfrac{2}{9} $ .
Hence, we see that line $ 2x + y - 4 = 0 $ divides line segment joining points $ A\left( {2, - 2} \right)\,\,and\,\,B\left( {3,7} \right) $ in ratio $ \dfrac{2}{9} $ .
So, the correct answer is “ $ \dfrac{2}{9} $”.

Note: For this type of problem we can either take the ratio as in terms of $ a:b $ and then use the section formula to find coordinates of the points which lie on both line segments joining given points and on a given line. Then substituting the coordinate of this point in the given line to form an equation in terms of a and b then finding the value of $ \dfrac{a}{b} $ by dividing the whole equation by b and hence required ratio asked in the given problem.