Last updated at March 11, 2017 by Teachoo

Transcript

Example 29 Show that the lines 𝑥 − 𝑎 + 𝑑𝛼 − 𝛿 = 𝑦 − 𝑎𝛼 = 𝑧 − 𝑎 − 𝑑𝛼 + 𝛿 and 𝑥 − 𝑏 + 𝑐𝛽 − 𝛾 = 𝑦 − 𝑏𝛽 = 𝑧 − 𝑏 − 𝑐𝛽 + 𝛾 are coplanar. Two lines 𝑥 − 𝑥1 𝑎1 = 𝑦 − 𝑦1 𝑏1 = 𝑧 − 𝑧1 𝑐1 and 𝑥 − 𝑥2 𝑎2 = 𝑦 − 𝑦2 𝑏2 = 𝑧 − 𝑧2 𝑐2 are coplanar if 𝒙𝟐− 𝒙𝟏 𝒚𝟐− 𝒚𝟏 𝒛𝟐− 𝒛𝟏 𝒂𝟏 𝒃𝟏 𝒄𝟏 𝒂𝟐 𝒃𝟐 𝒄𝟐 = 0 Now, 𝑥2− 𝑥1 𝑦2− 𝑦1 𝑧2− 𝑧1 𝑎1 𝑏1 𝑐1 𝑎2 𝑏2 𝑐2 = 𝑏−𝑐−𝑎 + 𝑑𝑏−𝑎𝑏+𝑐−𝑎−𝑑𝛼−𝛿𝛼𝛼+𝛿𝛽−𝛾𝛽𝛽+𝛾 Adding column 3 to column 1, = 2(𝑏−𝑎)𝑏−𝑎𝑏+𝑐−𝑎−𝑑2𝛼𝛼𝛼+𝛿2𝛽𝛽𝛽+𝛾 Taking 2 common from Column 1 = 2 𝑏 − 𝑎𝑏 − 𝑎𝑏 + 𝑐 − 𝑎 − 𝑑𝛼𝛼𝛼 + 𝛿𝛽𝛽𝛽 +𝛾 = 2 × 0 = 0 Therefore, the given two lines are coplanar .

Example 1

Example, 2

Example, 3 Important

Example, 4

Example, 5

Example, 6 Important

Example, 7

Example 8

Example, 9 Important

Example 10

Example 11

Example 12 Important

Example 13

Example 14

Example 15

Example 16

Example 17

Example 18

Example 19

Example 20 Important

Example 21 Important

Example 22

Example 23 Important

Example 24 Important

Example, 25 Important

Example 26

Example 27 Important

Example 28

Example 29 Important You are here

Example 30 Important

Chapter 11 Class 12 Three Dimensional Geometry

Serial order wise

About the Author

CA Maninder Singh

CA Maninder Singh is a Chartered Accountant for the past 8 years. He provides courses for Practical Accounts, Taxation and Efiling at teachoo.com .