Question 2 - Chapter 12 Class 11 Introduction to Three Dimensional Geometry (Important Question)
Last updated at April 16, 2024 by Teachoo
Chapter 12 Class 11 Introduction to Three Dimensional Geometry
Chapter 12 Class 11 Introduction to Three Dimensional Geometry
Last updated at April 16, 2024 by Teachoo
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) Question 2 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