Miscellaneous

Chapter 12 Class 11 - Intro to Three Dimensional Geometry
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Misc 5 A point R with x-coordinate 4 lies on the line segment joining the points P (2, β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),(β3)/(k + 1),(10k + 4)/(k + 1))] Given that Point R lie on the line segment PQ. and x β 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 ((γππ₯γ_2 + γππ₯γ_1)/(π + π),(γππ¦γ_2 + γππ¦γ_1)/(π + π),(γππ§γ_2 + γππ§γ_1)/(π + π)) Here, m = k , n = 1 x1 = 2 , y1 = β3 , z1 = 4 x2 = 8 , y2 = 0 , z2 = 10 Putting values R = ((π(8) + 1(2))/(π + 1),(π(0) + 1(β3))/(π + 1),(π(10) + 1(4))/(π + 1)) (4, b, c) = ((8π + 2)/(π + 1),(0 β 3)/(π + 1),(10π + 4)/(π + 1)) (4, b, c) = ((8π + 2)/(π + 1),(β3)/(π + 1),(10π + 4)/(π + 1)) x β coordinate 4 = (8π +2)/(π+1) 4(k + 1) = 8k + 2 4(k + 1 ) = (8 k + 2) 4k β 8k = 2 β 4 β 4k = β 2 k = (β2)/(β4) k = 1/2 y β Coordinate b = (β3)/(π+1) b(k + 1) = β 3 Putting k = 1/2 b (1/2+1) = β 3 b ((1 + 2)/2) = β 3 b (3/2) = β 3 b = (β3 Γ2)/3 b = β 2 z β Coordinate c = (10 π+4)/(π+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 Γ 2)/3 c = 6 Thus, a = 4 , b = -2 , c = 6 Hence, Coordinates of point R = (a ,b ,c) = (4, β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