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Chapter 12 Class 11 - Intro to Three Dimensional Geometry
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Misc 5 - A point R with x-coordinate 4 lies on line segment

Misc  5 - Chapter 12 Class 11 Introduction to Three Dimensional Geometry - Part 2
Misc  5 - Chapter 12 Class 11 Introduction to Three Dimensional Geometry - Part 3

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Misc 5 A point R with x-coordinate 4 lies on the line segment joining the points P (2, –3, 4) and Q (8, 0, 10). Find the coordinates of the point R. [Hint suppose R divides PQ in the ratio k: 1. The coordinates of the point R are given by ((8k + 2)/(k + 1),(−3)/(k + 1),(10k + 4)/(k + 1))] Given that Point R lie on the line segment PQ. and x – Coordinate of R is 4 Let Point R be (4, b, c) Let R divide line segment PR in the ratio k : 1 We know that Coordinate of Point that divides line in the ratio m : n is ((〖𝑚𝑥〗_2 + 〖𝑛𝑥〗_1)/(𝑚 + 𝑛),(〖𝑚𝑦〗_2 + 〖𝑛𝑦〗_1)/(𝑚 + 𝑛),(〖𝑚𝑧〗_2 + 〖𝑛𝑧〗_1)/(𝑚 + 𝑛)) Here, m = k , n = 1 x1 = 2 , y1 = –3 , z1 = 4 x2 = 8 , y2 = 0 , z2 = 10 Putting values R = ((𝑘(8) + 1(2))/(𝑘 + 1),(𝑘(0) + 1(−3))/(𝑘 + 1),(𝑘(10) + 1(4))/(𝑘 + 1)) (4, b, c) = ((8𝑘 + 2)/(𝑘 + 1),(0 − 3)/(𝑘 + 1),(10𝑘 + 4)/(𝑘 + 1)) (4, b, c) = ((8𝑘 + 2)/(𝑘 + 1),(−3)/(𝑘 + 1),(10𝑘 + 4)/(𝑘 + 1)) x – coordinate 4 = (8𝑘 +2)/(𝑘+1) 4(k + 1) = 8k + 2 4(k + 1 ) = (8 k + 2) 4k – 8k = 2 – 4 – 4k = – 2 k = (−2)/(−4) k = 1/2 y – Coordinate b = (−3)/(𝑘+1) b(k + 1) = – 3 Putting k = 1/2 b (1/2+1) = – 3 b ((1 + 2)/2) = – 3 b (3/2) = – 3 b = (−3 ×2)/3 b = – 2 z – Coordinate c = (10 𝑘+4)/(𝑘+1 ) c (k + 1 ) = 10k + 4 Putting k = 1/2 c (1/2+1) = 10 (1/2) + 4 c ((1+2)/2) = 5 + 4 c (3/2) = 9 c = (9 × 2)/3 c = 6 Thus, a = 4 , b = -2 , c = 6 Hence, Coordinates of point R = (a ,b ,c) = (4, –2, 6) Misc 5 A point R with x-coordinate 4 lies on the line segment joining the points P (2, 3, 4) and Q (8, 0, 10). Find the coordinates of the point R. [Hint suppose R divides PQ in the ratio k: 1. The coordinates of the point R are given by 8k+2 k+1 , 3 k+1 , 10k+4 k+1 ] Given that Point R lie on the line segment PQ. and x Coordinate of R is 4 Let Point R be (4, b, c) Let R divide line segment PR in the ratio k : 1 We know that Coordinate of Point that divide line segment joining (x1 y1 z1) & (x2 y2 z2) in the ratio m : n is 2 + 1 + , 2 + 1 + , 2 + 1 + Here, m = k , n = 1 x1 = 2 , y1 = 3 , x2 = 8 , y2 = 0 Putting values R = 8 +1(2) +1 , 0 +1( 3) +1 , 10 +1(4) +1 (4, b, c) = 8 +2 +1 , 0 3 +1 , 10 +4 +1 (4, b, c) = 8 +2 +1 , 3 +1 , 10 +4 +1

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.