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Example 29 - Chapter 11 Class 12 - Show lines are coplanar

Example 29 - Chapter 11 Class 12 Three Dimensional Geometry - Part 2
Example 29 - Chapter 11 Class 12 Three Dimensional Geometry - Part 3

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Example 29 Show that the lines (π‘₯ βˆ’ π‘Ž + 𝑑)/(𝛼 βˆ’ 𝛿) = (𝑦 βˆ’ π‘Ž)/𝛼 = (𝑧 βˆ’ π‘Ž βˆ’ 𝑑)/(𝛼 + 𝛿) nd (π‘₯ βˆ’ 𝑏 + 𝑐)/(𝛽 βˆ’ 𝛾) = (𝑦 βˆ’ 𝑏)/𝛽 = (𝑧 βˆ’ 𝑏 βˆ’ 𝑐)/(𝛽 + 𝛾) are coplanar.Two 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 are coplanar if |β– 8(𝒙_πŸβˆ’ 𝒙_𝟏&π’š_πŸβˆ’π’š_𝟏&𝒛_πŸβˆ’π’›_𝟏@𝒂_𝟏&𝒃_𝟏&𝒄_𝟏@𝒂_𝟐&𝒃_𝟐&𝒄_𝟐 )| = 0 (𝒙 βˆ’ 𝒂 + 𝒅)/(𝜢 βˆ’ 𝜹) = (π’š βˆ’ 𝒂)/𝜢 = (𝒛 βˆ’ 𝒂 βˆ’ 𝒅)/(𝜢 + 𝜹) (π‘₯ βˆ’ (π‘Ž βˆ’ 𝑑))/(𝛼 βˆ’ 𝛿) = (𝑦 βˆ’ π‘Ž)/𝛼 = (𝑧 βˆ’ (π‘Ž + 𝑑))/(𝛼 + 𝛿) Comparing (π‘₯ βˆ’ π‘₯_1)/π‘Ž_1 = (𝑦 βˆ’ 𝑦_1)/𝑏_1 = (𝑧 βˆ’ 𝑧_1)/𝑐_1 π‘₯_1 = π‘Ž βˆ’ d , 𝑦_1= π‘Ž , 𝑧_1= π‘Ž + d & π‘Ž_1=π›Όβˆ’π›Ώ, 𝑏_1= 𝛼, 𝑐_1= 𝛼+𝛿 (𝒙 βˆ’ 𝒃 + 𝒄)/(𝜷 βˆ’ 𝜸) = (π’š βˆ’ 𝒃)/𝜷 = (𝒛 βˆ’ 𝒃 βˆ’ 𝒄)/(𝜷 + 𝜸) (π‘₯ βˆ’ (𝑏 βˆ’ 𝑐))/(𝛽 βˆ’ 𝛾) = (𝑦 βˆ’ 𝑏)/𝛽 = (𝑧 βˆ’ (𝑏 + 𝑐))/(𝛽 + 𝛾) Comparing (π‘₯ βˆ’ π‘₯_2)/π‘Ž_2 = (𝑦 βˆ’ 𝑦_2)/𝑏_2 = (𝑧 βˆ’ 𝑧_1)/𝑐_2 π‘₯_2 = 𝑏 βˆ’ c , 𝑦_2= 𝑏 , 𝑧_2= 𝑏 + c & π‘Ž_2 = π›½βˆ’π›Ύ, 𝑏_2 = 𝛽, 𝑐_2 = 𝛽 + 𝛾 Now, |β– 8(π‘₯_2βˆ’π‘₯_1&𝑦_2βˆ’π‘¦_1&𝑧_2βˆ’π‘§_1@π‘Ž_1&𝑏_1&𝑐_1@π‘Ž_2&𝑏_2&𝑐_2 )| = |β– 8(π‘βˆ’π‘βˆ’π‘Ž + 𝑑&π‘βˆ’π‘Ž&𝑏+π‘βˆ’π‘Žβˆ’π‘‘@π›Όβˆ’π›Ώ&𝛼&𝛼+𝛿@π›½βˆ’π›Ύ&𝛽&𝛽+𝛾)| Adding Column 3 to Column 1, = |β– 8(π‘βˆ’π‘βˆ’π‘Ž + 𝑑+(𝑏+π‘βˆ’π‘Žβˆ’π‘‘)&π‘βˆ’π‘Ž&𝑏+π‘βˆ’π‘Žβˆ’π‘‘@π›Όβˆ’π›Ώ+(𝛼+𝛿)&𝛼&𝛼+𝛿@π›½βˆ’π›Ύ+(𝛽+𝛾)&𝛽&𝛽+𝛾)| = |β– 8(2(π‘βˆ’π‘Ž)&π‘βˆ’π‘Ž&𝑏+π‘βˆ’π‘Žβˆ’π‘‘@2𝛼&𝛼&𝛼+𝛿@2𝛽&𝛽&𝛽+𝛾)| Taking 2 common from Column 1 = 2 |β– 8(𝑏 βˆ’ π‘Ž&𝑏 βˆ’ π‘Ž&𝑏 + 𝑐 βˆ’ π‘Ž βˆ’ 𝑑@𝛼&𝛼&𝛼 + 𝛿@𝛽&𝛽&𝛽 +𝛾)| = 2 Γ— 0 = 0 Therefore, the given two lines are coplanar. Since Columns 1 and 2 are same, The value of determinant is zero.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.