Question

The coordinate of the point which divides the line segment joining points $ A\left( {0,0} \right) $ and $ B\left( {9,12} \right) $ in the ratio $ 1:2 $ are:
A. $ \left( { - 3,4} \right) $
B. $ \left( {3,4} \right) $
C. $ \left( {3, - 4} \right) $
D.None of these

Answer 1

Hint: Use the section formula to find the coordinates of the point that divides the line segments joining the two points in the given ratio or in other sense the point divides the line segment into two parts one with double the distance of the other.

Complete step-by-step answer:
The end points of the line segment are $ A\left( {0,0} \right) $ and $ B\left( {9,12} \right) $ , the ratio is $ 1:2 $ .
The formula for the coordinates of the point that divide the line segment joining the points $ \left( {{x_1},{y_1}} \right) $ and $ \left( {{x_2},{y_2}} \right) $ in the $ m:n $ ratio is equal to $ \left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right) $ .
As per given in the question the endpoints of the line segment are $ A\left( {0,0} \right) $ and $ B\left( {9,12} \right) $ and the given ratio is $ 1:2 $ .
So, the value of $ m $ and $ n $ is equal to $ 1 $ and $ 2 $ respectively.
Substitute the end points and ratio in the formula for coordinates of the point:
 $
  \left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)
   = \left( {\dfrac{{1 \times 9 + 2 \times 0}}{{1 + 2}},\dfrac{{1 \times 12 + 2 \times 0}}{{1 + 2}}} \right) \\
   = \left( {\dfrac{9}{3},\dfrac{{12}}{3}} \right) \\
   = \left( {3,4} \right) \;
  $
So, the coordinates of the point which divides the line segment joining points $ A\left( {0,0} \right) $ and $ B\left( {9,12} \right) $ in the ratio $ 1:2 $ are $ \left( {3,4} \right) $ .
One can easily verify that this point cuts the line into two parts such that the length of one is double the other.
So, the correct answer is “Option B”.

Note: The formula for the coordinates of the point that divide the line segment joining the points $ \left( {{x_1},{y_1}} \right) $ and $ \left( {{x_2},{y_2}} \right) $ in the $ m:n $ ratio is equal to $ \left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right) $ . As this line passes through origin so this can be found by using the equation of the line too.