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Example 7 - Find coordinates of point which divides line - Section - Defination

  1. Chapter 12 Class 11 Introduction to Three Dimensional Geometry
  2. Serial order wise
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Example 7 Find the coordinates of the point which divides the line segment joining the points (1, –2, 3) and (3, 4, –5) in the ratio 2 : 3 (i) internally, Let the 2 given points be A (1, 2, 3) & B (3, 4, – 5) Let P (x, y, z,) be points that divides line in ratio 2:3 internally We know that Co-ordinate of point P (x, y, z) that divides the line segment joining A (x1, y1, z1) & B (x2, y2, z2) internally in the ratio m : n is P (x, y, z,) = ﷐﷐﷐𝑚 𝑥﷮2﷯+﷐ 𝑛 𝑥﷮1﷯﷮𝑚+𝑛﷯,﷐﷐𝑚 𝑦﷮2﷯+﷐ 𝑛 𝑦﷮1﷯﷮𝑚+𝑛﷯,﷐﷐𝑚 𝑧﷮2﷯+﷐ 𝑛 𝑧﷮1﷯﷮𝑚+𝑛﷯﷯ Here, x1 = 1, y1 = – 2, z1 = 3 x2 = 3, y2 = 4, z2 = – 5 & m = 2, n = 3 Putting values Co-ordinate of point P of (x, y, z) = ﷐﷐2(3) + 3﷐1﷯﷮2+3﷯,﷐2﷐4﷯ + 3﷐−2﷯﷮2 + 3﷯,﷐2﷐−5﷯ + 3﷐3﷯﷮2 + 3﷯﷯ = ﷐﷐6 + 3﷮5﷯,﷐8 − 6﷮5﷯,﷐−10 + 9﷮5﷯﷯ = ﷐﷐9﷮5﷯,﷐2﷮5﷯,﷐−1﷮5﷯﷯ Thus, the required co-ordinate is ﷐﷐9﷮5﷯,﷐2﷮5﷯,﷐−1﷮5﷯﷯ Example 7 Find the coordinates of the point which divides the line segment joining the points (1, –2, 3) and (3, 4, –5) in the ratio 2 : 3 (ii) externally, Let the 2 given points be A (1, 2, 3) & B (3, 4, – 5) Let P (x, y, z,) be points that divides line in ratio 2:3 internally We know that Co-ordinate of point P (x, y, z) that divides the line segment joining A (x1, y1, z1) & B (x2, y2, z2) externally in the ratio m : n is P (x, y, z,) = ﷐﷐﷐𝑚 𝑥﷮2﷯−﷐ 𝑛 𝑥﷮1﷯﷮𝑚−𝑛﷯,﷐﷐𝑚 𝑦﷮2﷯−﷐ 𝑛 𝑦﷮1﷯﷮𝑚−𝑛﷯,﷐﷐𝑚 𝑧﷮2﷯−﷐ 𝑛 𝑧﷮1﷯﷮𝑚−𝑛﷯﷯ Here, x1 = 1, y1 = – 2, z1 = 3 x2 = 3, y2 = 4, z2 = – 5 & m = 2, n = 3 Putting values Co-ordinate of point P is (x, y, z) = ﷐﷐2(3) − 3﷐1﷯﷮2−3﷯,﷐2﷐4﷯ − 3﷐−2﷯﷮2 − 3﷯,﷐2﷐−5﷯ − 3﷐3﷯﷮2 − 3﷯﷯ = ﷐﷐6 − 3﷮(−1)﷯,﷐8 + 6﷮(−1)﷯,﷐−10 − 9﷮(−1)﷯﷯ = ﷐﷐3﷮(−1)﷯,﷐14﷮(−1)﷯,﷐ −19﷮(−1)﷯﷯ = ( – 3 , – 14 , 19) Thus, the required co-ordinate is (– 3 , – 14 , 19)

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