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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Question 19 Show that the lines (𝑥 − 𝑎 + 𝑑)/(𝛼 − 𝛿) = (𝑦 − 𝑎)/𝛼 = (𝑧 − 𝑎 − 𝑑)/(𝛼 + 𝛿) nd (𝑥 − 𝑏 + 𝑐)/(𝛽 − 𝛾) = (𝑦 − 𝑏)/𝛽 = (𝑧 − 𝑏 − 𝑐)/(𝛽 + 𝛾) are coplanar.Two lines (𝑥 − 𝑥_1)/𝑎_1 = (𝑦 − 𝑦_1)/𝑏_1 = (𝑧 − 𝑧_1)/𝑐_1 and (𝑥 − 𝑥_2)/𝑎_2 = (𝑦 − 𝑦_2)/𝑏_2 = (𝑧 − 𝑧_2)/𝑐_2 are coplanar if |■8(𝒙_𝟐− 𝒙_𝟏&𝒚_𝟐−𝒚_𝟏&𝒛_𝟐−𝒛_𝟏@𝒂_𝟏&𝒃_𝟏&𝒄_𝟏@𝒂_𝟐&𝒃_𝟐&𝒄_𝟐 )| = 0 (𝒙 − 𝒂 + 𝒅)/(𝜶 − 𝜹) = (𝒚 − 𝒂)/𝜶 = (𝒛 − 𝒂 − 𝒅)/(𝜶 + 𝜹) (𝑥 − (𝑎 − 𝑑))/(𝛼 − 𝛿) = (𝑦 − 𝑎)/𝛼 = (𝑧 − (𝑎 + 𝑑))/(𝛼 + 𝛿) Comparing (𝑥 − 𝑥_1)/𝑎_1 = (𝑦 − 𝑦_1)/𝑏_1 = (𝑧 − 𝑧_1)/𝑐_1 𝑥_1 = 𝑎 − d , 𝑦_1= 𝑎 , 𝑧_1= 𝑎 + d & 𝑎_1=𝛼−𝛿, 𝑏_1= 𝛼, 𝑐_1= 𝛼+𝛿 (𝒙 − 𝒃 + 𝒄)/(𝜷 − 𝜸) = (𝒚 − 𝒃)/𝜷 = (𝒛 − 𝒃 − 𝒄)/(𝜷 + 𝜸) (𝑥 − (𝑏 − 𝑐))/(𝛽 − 𝛾) = (𝑦 − 𝑏)/𝛽 = (𝑧 − (𝑏 + 𝑐))/(𝛽 + 𝛾) Comparing (𝑥 − 𝑥_2)/𝑎_2 = (𝑦 − 𝑦_2)/𝑏_2 = (𝑧 − 𝑧_1)/𝑐_2 𝑥_2 = 𝑏 − c , 𝑦_2= 𝑏 , 𝑧_2= 𝑏 + c & 𝑎_2 = 𝛽−𝛾, 𝑏_2 = 𝛽, 𝑐_2 = 𝛽 + 𝛾 Now, |■8(𝑥_2−𝑥_1&𝑦_2−𝑦_1&𝑧_2−𝑧_1@𝑎_1&𝑏_1&𝑐_1@𝑎_2&𝑏_2&𝑐_2 )| = |■8(𝑏−𝑐−𝑎 + 𝑑&𝑏−𝑎&𝑏+𝑐−𝑎−𝑑@𝛼−𝛿&𝛼&𝛼+𝛿@𝛽−𝛾&𝛽&𝛽+𝛾)| Adding Column 3 to Column 1, = |■8(𝑏−𝑐−𝑎 + 𝑑+(𝑏+𝑐−𝑎−𝑑)&𝑏−𝑎&𝑏+𝑐−𝑎−𝑑@𝛼−𝛿+(𝛼+𝛿)&𝛼&𝛼+𝛿@𝛽−𝛾+(𝛽+𝛾)&𝛽&𝛽+𝛾)| = |■8(2(𝑏−𝑎)&𝑏−𝑎&𝑏+𝑐−𝑎−𝑑@2𝛼&𝛼&𝛼+𝛿@2𝛽&𝛽&𝛽+𝛾)| Taking 2 common from Column 1 = 2 |■8(𝑏 − 𝑎&𝑏 − 𝑎&𝑏 + 𝑐 − 𝑎 − 𝑑@𝛼&𝛼&𝛼 + 𝛿@𝛽&𝛽&𝛽 +𝛾)| = 2 × 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 14 years. He provides courses for Maths, Science and Computer Science at Teachoo