Examples

Example 1

Example, 2 Important

Example, 3

Example, 4 Important

Example, 5 Important

Example, 6 Important

Example, 7

Example 8 Important

Example 9

Example 10 Important

Question 1

Question 2

Question 3 Important

Question 4

Question 5

Question 6 Important

Question 7

Question 8

Question 9 Important

Question 10 Important

Question 11 Important

Question 12

Question 13 Important

Question 14

Question 15 Important

Question 16

Question 17 Important

Question 18 Important

Question 19 Important You are here

Question 20 Important

Chapter 11 Class 12 Three Dimensional Geometry

Serial order wise

Last updated at April 16, 2024 by Teachoo

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.