Example 10 - Find ratio in which line joining (4, 8, 10) - Examples

  1. Chapter 12 Class 11 Introduction to Three Dimensional Geometry
  2. Serial order wise
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Example, 10 Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, – 8) is divided by the YZ-plane. Let AB be the line segment joining points A (4, 8, 10) & (6, 10, – 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﷮𝑚 + 𝑛﷯,﷐𝑚𝑧2 + 𝑛𝑧1﷮𝑚 + 𝑛﷯﷯ Here, m = k , n = 1 x1 = 4, y1 = 8, z1 = 10 x2 = 6 , y2 = 10, z2 = – 8 Co- ordinate of P P (x, y , z) = ﷐﷐𝑘 ﷐6﷯ + 1 ﷐4﷯﷮𝑘 + 1﷯, ﷐𝑘﷐10﷯ + 1﷐8﷯﷮𝑘 + 1﷯, ﷐𝑘﷐−8﷯ + 1﷐10﷯﷮𝑘 + 1﷯﷯ (x, y , z) = ﷐﷐6𝑘 + 4﷮k + 1﷯,﷐10k + 8﷮k + 1﷯,﷐−8𝑘 + 10﷮k + 1﷯﷯ Since Point p (x, y, z) lie on the YZ plane its x – coordinate will be zero P (0, y , z) = ﷐﷐6𝑘 + 4﷮k + 1﷯,﷐10k + 8﷮k + 1﷯,﷐−8𝑘 + 10﷮k + 1﷯﷯ Comparing x – Coordinate 0 = ﷐6𝑘 + 4﷮𝑘 + 1﷯ (k + 1) (0) = 6k + 4 0 = 6k + 4 6k + 4 = 0 6k = – 4 k = ﷐−4﷮6﷯ k = ﷐−2﷮3﷯ ﷐𝑘﷮1﷯ = ﷐−2﷮3﷯ k : 1 = – 2 : 3 Since value of k is negative Hence YZ Plane divide AB externally in the ratio 2 : 3

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