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 Ex 12.3, 3 - Find ratio in which YZ-plane divides 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, 3 Find the ratio in which the YZ-plane divides the line segment formed by joining the points (–2, 4, 7) and (3, –5, 8). Let AB be the line segment joining points A (–2, 4, 7) & B (3, –5, 8) Let YZ Plane divide line AB at P (x, y, z) in the ratio k : 1 Co-ordinate of P that divide line segment joining point A (x1, y1, z1) & B((x2, y2, z2) in the ratio m : n is = ﷐﷐𝑚𝑥2+𝑛𝑥1﷮𝑚+𝑛﷯, ﷐𝑚𝑦2+𝑛𝑦1﷮𝑚2+𝑛﷯,﷐𝑚𝑧2+𝑛𝑧1﷮𝑚+𝑛﷯﷯ Here, m = k , n = 1 x1 = – 2, y1 = 4, z1 = 7 x2 = 3, y2 = – 5, z2 = 8 Co- ordinate of P P (x, y , z) = ﷐﷐𝑘 ﷐3﷯ + 1 ﷐−2﷯﷮𝑘 + 1﷯, ﷐𝑘﷐−5﷯ + 1﷐4﷯﷮𝑘 + 1﷯, ﷐𝑘﷐8﷯ + 1﷐7﷯﷮𝑘 + 1﷯﷯ (x, y , z) = ﷐﷐3𝑘 − 2﷮k + 1﷯,﷐−5𝑘 + 4﷮k + 1﷯,﷐8𝑘 + 7﷮k + 1﷯﷯ Since Point p (x, y, z) lie on the YZ plane its x – coordinate will be zero P(0, y , z) = ﷐﷐3𝑘 − 2﷮k + 1﷯,﷐−5𝑘 + 4﷮k + 1﷯,﷐8𝑘 + 7﷮k + 1﷯﷯ Comparing x – Co- ordinate 0 = ﷐3𝑘 − 2﷮𝑘 + 1﷯ (k + 1) (0) = 3k – 2 0 = 3k – 2 3k – 2 = 0 3k = 2 k = ﷐2﷮3﷯ ﷐𝑘﷮1﷯ = ﷐2﷮3﷯ k : 1 = 2 : 3 Thus, YZ plane divides AB in the ratio 2 : 3

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