Answer Hint: In the given problem, we are asked to find the ratio and co-ordinates of the point of division such that the line $ 2x + 3y - 5 = 0 $ divides the line segment joining the points $ \left( {8, - 9} \right) $ and $ \left( {2,1} \right) $ . In order to find the ratio we will use a section formula. Complete step by step solution: In this given problem, the line of equation is $ 2x + 3y - 5 = 0 $ . Let $ A\left( {{x_1},{y_1}} \right) = \left( {8, - 9} \right) $ and $ B\left( {{x_2},{y_2}} \right) = \left( {2,1} \right) $ be the end points of line segment $ AB $ and $ P\left( {x,y} \right) $ be the point of division of line segment joining the $ AB $ .
So, the correct answer is “ $ \left( {\dfrac{{24}}{9},\dfrac{{ - 1}}{9}} \right) $ .” Note: According to the section formula, the coordinates of the point $ P\left( {x,y} \right) $ which divides the line segment joining the point $ A\left( {{x_1},{y_1}} \right) $ and $ B\left( {{x_2},{y_2}} \right) $ in the ratio $ {m_1}:{m_2} $ internally, is $ \left( {\dfrac{{{m_1}{x_1} + {m_2}{x_2}}}{{{m_1} + {m_2}}},\dfrac{{{m_1}{y_1} + {m_2}{y_2}}}{{{m_1} + {m_2}}}} \right) $ . Then by using this ratio we will find the coordinates of the point of division. If the midpoint of a line segment divides the line segment in the ratio $ 1:1 $ then the coordinates of the mid-point $ P $ is $ \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right) $ . Click here to get PDF DOWNLOAD for all questions and answers of this Book - NAGEEN PRAKASHAN Class 10 MATHS
1) 1 : 2
2) 2 : 1
3) 2 : 3
4) 3 : 4
Solution:
Let the line y – x + 2 = 0 divide the line joining the points A(3, -1) and B(8, 9) in the ratio k:1. Let C(x, y) be the point of intersection of these two lines. Then by section formula x = (mx2+nx1)/(m+n) y = (my2+ny1)/(m+n) Here (x1, y1) = (3, -1) (x2, y2) = (8, 9) m:n = k:1 So x = (8k + 3)/(k+1) y = (9k – 1)/(k+1) The point C lies on the line y – x + 2 = 0 So (9k – 1)/(k+1) – (8k + 3)/(k+1) + 2 = 0 => (9k – 1 – 8k – 3 + 2k + 2)/(k+1) = 0 => (3k – 2)/(k+1) = 0 => 3k – 2 = 0 => k = 2/3 So the required ratio is 2:3. Hence option (3) is the answer.
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