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  1. Chapter 11 Class 12 Three Dimensional Geometry
  2. Serial order wise

Transcript

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 = (๐‘จ_๐Ÿ ๐‘จ_๐Ÿ + ๐‘ฉ_๐Ÿ ๐‘ฉ_๐Ÿ + ๐‘ช_๐Ÿ ๐‘ช_๐Ÿ)/(โˆš(ใ€–๐‘จ_๐Ÿใ€—^๐Ÿ + ใ€–๐‘ฉ_๐Ÿใ€—^๐Ÿ + ใ€–๐‘ช_๐Ÿใ€—^๐Ÿ ) โˆš(ใ€–๐‘จ_๐Ÿใ€—^๐Ÿ + ใ€–๐‘ฉ_๐Ÿใ€—^๐Ÿ + ใ€–๐‘ช_๐Ÿใ€—^๐Ÿ )) Given the two planes are 2x โˆ’ 1y + 4z = 5 Comparing with A1x + B1y + C1z = d1 A1 = 2 , B1 = โ€“1 , C1 = 4 , ๐‘‘_1= 5 5x โˆ’ 2.5y + 10z = 6 Multiplying by 2 on both sides, 10x โˆ’ 5y + 20z = 12 Comparing with A2x + B2y + C2z = d2 A2 = 10 , B2 = โ€“5 , C2 = 20 , ๐‘‘_2= 12 So, cos ๐œƒ = |((2 ร— 10) + (โˆ’1 ร— โˆ’5) + (4 ร— 20))/(โˆš(2^2 + (ใ€–โˆ’1)ใ€—^2 + 4^2 ) โˆš(ใ€–10ใ€—^(2 )+ (ใ€–โˆ’5)ใ€—^2 + ใ€–20ใ€—^2 ))| = |(20 + 5 + 80)/(โˆš(4 + 1 + 16) โˆš(100 + 25 + 400))| = |105/(โˆš21 โˆš525)| = |105/(โˆš21 ร— โˆš(25 ร— 21))| = |105/(โˆš21 ร— 5 โˆš21)| = |105/(21 ร— 5)| = 1 So, cos ฮธ = 1 โˆด 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 โˆ’ 1y + 4z = 5 Comparing with A1x + B1y + C1z = d1 Direction ratios of normal = 2, โ€“1, 4 A1 = 2 , B1 = โ€“1 , C1 = 4 5x โˆ’ 2.5y + 10z = 6 Multiplying by 2 on both sides, 10x โˆ’ 5y + 20z = 12 Comparing with A2x + B2y + C2z = d2 Direction ratios of normal = 10, โ€“5, 20 A2 = 10 , B2 = โ€“5 , C2 = 20 Two lines are parallel if their direction ratios are proportional. ๐ด_1/๐ด_2 = 2/10 = 1/5 , ๐ต_1/๐ต_2 = (โˆ’1)/(โˆ’5) = 1/5 , ๐ถ_1/๐ถ_2 = 4/20 = 1/5 a Since, ๐‘จ_๐Ÿ/๐‘จ_๐Ÿ = ๐‘ฉ_๐Ÿ/๐‘ฉ_๐Ÿ = ๐‘ช_๐Ÿ/๐‘ช_๐Ÿ = ๐Ÿ/๐Ÿ“ Therefore, the normal vectors of the two planes are parallel. So, the two planes are parallel. So, option (B) is correct

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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.