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Class 12
Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Find the acute angle between the lines

(x - 4)/3 = (y + 3)/4 = (z + 1)/5 and (x - 1)/4 = (y + 1)/(-3) = (z + 10)/5

Find the acute angle between lines  (x - 4)/3 = (y + 3)/4 = (z + 1)/5

Question 25 - CBSE Class 12 Sample Paper for 2020 Boards - Part 2
Question 25 - CBSE Class 12 Sample Paper for 2020 Boards - Part 3


Transcript

Question 25 Find the acute angle between the lines (π‘₯ βˆ’ 4)/3 = (𝑦 + 3)/4 = (𝑧 + 1)/5 and (π‘₯ βˆ’ 1)/4 = (𝑦 + 1)/(βˆ’3) = (𝑧 + 10)/5 Angle between the pair of lines (π‘₯ βˆ’ π‘₯_1)/π‘Ž_1 = (𝑦 βˆ’ 𝑦_1)/𝑏_1 = (𝑧 βˆ’ 𝑧_1)/𝑐_1 and (π‘₯ βˆ’ π‘₯_2)/π‘Ž_2 = (𝑦 βˆ’ 𝑦_2)/𝑏_2 = (𝑧 βˆ’ 𝑧_2)/𝑐_2 is given by cos ΞΈ = |(π‘Ž_1 π‘Ž_2 + 𝑏_1 𝑏_2 + 𝑐_1 𝑐_2)/(√(γ€–π‘Ž_1γ€—^2 + 〖𝑏_1γ€—^2 + 〖𝑐_1γ€—^2 ) √(γ€–π‘Ž_2γ€—^2 + 〖𝑏_2γ€—^2 + 〖𝑐_2γ€—^2 ))| (𝒙 βˆ’ πŸ’)/πŸ‘ = (π’š + πŸ‘)/πŸ’ = (𝒛 + 𝟏)/πŸ“ Comparing with (π‘₯ βˆ’ π‘₯_1)/π‘Ž_1 = (𝑦 βˆ’ 𝑦_1)/𝑏_1 = (𝑧 βˆ’ 𝑧_1)/𝑐_1 π‘Ž1 = 3, b1 = 4, c1 = 4 (𝒙 βˆ’ 𝟏)/πŸ’ = (π’š + 𝟏)/(βˆ’πŸ‘) = (𝒛 + 𝟏𝟎)/πŸ“ Comparing with (π‘₯ βˆ’ π‘₯_2)/π‘Ž_2 = (𝑦 βˆ’ 𝑦_2)/𝑏_2 = (𝑧 βˆ’ 𝑧_2)/𝑐_2 π‘Ž2 = 4, 𝑏2 = –3, 𝑐2 = 5 Now, cos ΞΈ = |(π‘Ž_1 π‘Ž_2 + 𝑏_1 𝑏_2 + 𝑐_1 𝑐_2)/(√(γ€–π‘Ž_1γ€—^2 + 〖𝑏_1γ€—^2 + 〖𝑐_1γ€—^2 ) √(γ€–π‘Ž_2γ€—^2 + 〖𝑏_2γ€—^2 + 〖𝑐_2γ€—^2 ))| = |(3 Γ— 4 + 4 Γ— (βˆ’3) + 5 Γ— 5)/(√(3^2 + 4^2 + 5^2 ) √(4^2 +(βˆ’3)^2 + 5^2 ))| = |(12 βˆ’ 12 + 25)/(√(9 + 16 + 25) √(16 + 9 + 25))| = |25/(√50 √50)| = |25/50| = |1/2| = 1/2 So, cos ΞΈ = 1/2 ∴ ΞΈ = 60Β° = 𝝅/πŸ‘ Therefore, required angle is 𝝅/πŸ‘ Note: Please write angle in radians and not degree

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.