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Last updated at Feb. 1, 2020 by Teachoo

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Misc 23 (Method 1) The planes: 2x โ y + 4z = 5 and 5x โ 2.5y + 10z = 6 are (A) Perpendicular (B) Parallel (C) intersect y-axis (D) passes through (0,0, 5/4) Angle between two planes A1x + B1y + C1z = d1 and A2x + B2y + C2z = d2 is given by cos ฮธ = (๐จ_๐ ๐จ_๐ + ๐ฉ_๐ ๐ฉ_๐ + ๐ช_๐ ๐ช_๐)/(โ(ใ๐จ_๐ใ^๐ + ใ๐ฉ_๐ใ^๐ + ใ๐ช_๐ใ^๐ ) โ(ใ๐จ_๐ใ^๐ + ใ๐ฉ_๐ใ^๐ + ใ๐ช_๐ใ^๐ )) Given the two planes are 2x โ 1y + 4z = 5 Comparing with A1x + B1y + C1z = d1 A1 = 2 , B1 = โ1 , C1 = 4 , ๐_1= 5 5x โ 2.5y + 10z = 6 Multiplying by 2 on both sides, 10x โ 5y + 20z = 12 Comparing with A2x + B2y + C2z = d2 A2 = 10 , B2 = โ5 , C2 = 20 , ๐_2= 12 So, cos ๐ = |((2 ร 10) + (โ1 ร โ5) + (4 ร 20))/(โ(2^2 + (ใโ1)ใ^2 + 4^2 ) โ(ใ10ใ^(2 )+ (ใโ5)ใ^2 + ใ20ใ^2 ))| = |(20 + 5 + 80)/(โ(4 + 1 + 16) โ(100 + 25 + 400))| = |105/(โ21 โ525)| = |105/(โ21 ร โ(25 ร 21))| = |105/(โ21 ร 5 โ21)| = |105/(21 ร 5)| = 1 So, cos ฮธ = 1 โด ฮธ = 0ยฐ Since angle between the planes is 0ยฐ, Therefore, the planes are parallel. So, Option (B) is correct Misc 23 (Method 2) The planes: 2x โ y + 4z = 5 and 5x โ 2.5y + 10z = 6 are (A) Perpendicular (B) Parallel (C) intersect y-axis (D) passes through (0,0, 5/4) 2x โ 1y + 4z = 5 Comparing with A1x + B1y + C1z = d1 Direction ratios of normal = 2, โ1, 4 A1 = 2 , B1 = โ1 , C1 = 4 5x โ 2.5y + 10z = 6 Multiplying by 2 on both sides, 10x โ 5y + 20z = 12 Comparing with A2x + B2y + C2z = d2 Direction ratios of normal = 10, โ5, 20 A2 = 10 , B2 = โ5 , C2 = 20 Two lines are parallel if their direction ratios are proportional. ๐ด_1/๐ด_2 = 2/10 = 1/5 , ๐ต_1/๐ต_2 = (โ1)/(โ5) = 1/5 , ๐ถ_1/๐ถ_2 = 4/20 = 1/5 a Since, ๐จ_๐/๐จ_๐ = ๐ฉ_๐/๐ฉ_๐ = ๐ช_๐/๐ช_๐ = ๐/๐ Therefore, the normal vectors of the two planes are parallel. So, the two planes are parallel. So, option (B) is correct

Miscellaneous

Misc 1
Important

Misc 2

Misc 3

Misc 4 Important

Misc 5 Important

Misc 6 Important

Misc 7

Misc 8 Important

Misc 9 Important

Misc 10

Misc 11 Important

Misc 12 Important

Misc 13 Important

Misc 14 Important

Misc 15 Important

Misc 16 Important

Misc 17 Important

Misc 18 Important

Misc 19 Important

Misc 20 Important

Misc 21 Important

Misc 22 Important

Misc 23 Important You are here

Chapter 11 Class 12 Three Dimensional Geometry

Serial order wise

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.