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Ex 12.3, 5 - Find coordinates of points which trisect line - Ex 12.3

  1. Chapter 12 Class 11 Introduction to Three Dimensional Geometry
  2. Serial order wise
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Ex 12.3, 5 (Method 1) Find the coordinates of the points which trisect the line segment joining the points P (4, 2, –6) and Q (10, –16, 6). Let Point A (a, b, c) & point B (p, q, r) trisect the line segment PQ i.e. PA = AB = BC Point A divides PQ in the ratio of 1 : 2 We know that , Coordinate of point 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, m = 1 , n = 2 x1 = 4 , y1 = 2 , z1 = – 6 x2 = 10 , y2 = – 16 , z2 = 6 Coordinate of A are (a, b, c) = ﷐﷐10 ﷐1﷯ + 4 ﷐2﷯﷮1 + 2﷯,﷐−16 ﷐1﷯ + 2 ﷐2﷯﷮1 + 2﷯,﷐6 ﷐1﷯ + ﷐− 6﷯ ﷐2﷯﷮1 + 2﷯﷯ (a, b, c) = ﷐﷐10 + 8﷮3﷯,﷐− 16 + 4﷮3﷯,﷐6 − 12﷮3﷯﷯ (a, b, c) = ( 6, – 4, –2) Hence coordinate of A = (6, – 4, –2) Similarly Point B (p, q, r) divide line PQ in the ratio 2 : 1 We know that coordinate of point that divide line segment joining (x1 y1 z1) , ( x2 , y2 z2) in the ration m : n are = ﷐﷐﷐𝑚𝑥﷮2﷯+﷐𝑛𝑥﷮1﷯﷮𝑚+𝑛﷯,﷐﷐𝑚𝑦﷮2﷯+﷐𝑛𝑦﷮1﷯﷮𝑚+𝑛﷯,﷐﷐𝑚𝑧﷮2﷯+﷐𝑛𝑧﷮1﷯﷮𝑚+𝑛﷯﷯ Here, m = 2 , n = 1 x1 = 4 , y1 = 2 , z1 = – 6 x2 = 10 , y2 = – 16 , z2 = 6 Coordinate of B are (p, q, r) = ﷐﷐10 ﷐2﷯ + 4 ﷐1﷯﷮2 + 1﷯,﷐−16 ﷐2﷯ + 2 ﷐1﷯﷮2 + 1﷯,﷐6 ﷐2﷯ + ﷐− 6﷯ ﷐1﷯﷮2 + 1﷯﷯ (p, q, r) = ﷐﷐20 + 4﷮2 + 1﷯,﷐− 32 + 2﷮2 + 1﷯,﷐12 − 6 ﷮2 + 1﷯﷯ (p, q, r) = ﷐﷐24﷮3﷯,﷐− 30﷮3﷯,﷐6﷮3﷯﷯ (p, q, r) = ( 8, – 10, 2) Hence coordinate of Point B = (6, – 10, 2) Ex 12.3, 5 (Method 2) Find the coordinates of the points which trisect the line segment joining the points P (4, 2, –6) and Q (10, –16, 6). Let Point A (a, b, c) & point B (p, q, r) trisect the line segment PQ i.e. PA = AB = BC Point A divides PQ in the ratio of 1 : 2 We know that , Coordinate of point 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, m = 1 , n = 2 x1 = 4 , y1 = 2 , z1 = – 6 x2 = 10 , y2 = – 16 , z2 = 6 Coordinate of A are (a, b, c) = ﷐﷐10 ﷐1﷯ + 4 ﷐2﷯﷮1 + 2﷯,﷐−16 ﷐1﷯ + 2 ﷐2﷯﷮1 + 2﷯,﷐6 ﷐1﷯ + ﷐− 6﷯ ﷐2﷯﷮1 + 2﷯﷯ (a, b, c) = ﷐﷐10 + 8﷮3﷯,﷐− 16 + 4﷮3﷯,﷐6 − 12﷮3﷯﷯ (a, b, c) = ( 6, – 4, –2) Hence coordinate of A = (6, – 4, –2) Now, Point B (p, q, r) is the mid-point of line AQ Coordinate of point B which is mid point of line AQ A = (6, – 4, – 2) Q = (10, – 16, 6) Here, x1 = 6 , y1 = – 4 , z1 = – 2 x2 = 10 , y2 = – 16 , z2 = 6 Coordinate of B are = ﷐﷐6 + 10﷮2﷯,﷐− 4 + ( − 16)﷮2﷯,﷐− 2 + 6﷮2﷯﷯ = ﷐﷐16﷮2﷯,﷐− 20﷮2﷯,﷐4﷮2﷯﷯ = ( 8, – 10, 2) Thus point A (6, – 4, 2) & B (8, – 10, 2) trisect the line segment PQ.

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