Check sibling questions

Example 21 - Show that lines x+3/3 = y-1/1 = z-5/5 are coplanar

Example 21 - Chapter 11 Class 12 Three Dimensional Geometry - Part 2
Example 21 - Chapter 11 Class 12 Three Dimensional Geometry - Part 3


Transcript

Example 21 Show that the lines (π‘₯ + 3)/( βˆ’3) = (𝑦 βˆ’ 1)/1 = (𝑧 βˆ’ 5)/5 and (π‘₯ + 1)/( βˆ’1) = (𝑦 βˆ’ 2)/2 = (𝑧 βˆ’ 5)/5 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’π’™_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 Given, the two lines are Given, (𝒙 + πŸ‘)/( βˆ’ πŸ‘) = (π’š βˆ’ 𝟏)/𝟏 = (𝒛 βˆ’ πŸ“)/πŸ“ (π‘₯ βˆ’(βˆ’3))/( βˆ’ 3) = (𝑦 βˆ’1)/1 = (π‘§βˆ’ 5)/5 Comparing (π‘₯ βˆ’ π‘₯_1)/π‘Ž_1 = (𝑦 βˆ’ 𝑦_1)/𝑏_1 = (𝑧 βˆ’ 𝑧_1)/𝑐_1 π‘₯_1 = βˆ’3, 𝑦_1 = 1, 𝑧_1= 5 & π‘Ž_1 = βˆ’3, 𝑏_1 = 1, 𝑐_1= 5 Given, (𝒙 + 𝟏)/( βˆ’ 𝟏) = (π’š βˆ’ 𝟐)/𝟏 = (𝒛 βˆ’ πŸ“)/πŸ“ (π‘₯ βˆ’ (βˆ’1))/( βˆ’ 1) = (𝑦 βˆ’ 2)/2 = (𝑧 βˆ’ 5)/5 Comparing (π‘₯ βˆ’ π‘₯_2)/π‘Ž_2 = (𝑦 βˆ’ 𝑦_2)/𝑏_2 = (𝑧 βˆ’ 𝑧_2)/𝑐_2 π‘₯_2 = βˆ’1, 𝑦_2 = 2, 𝑧_2= 5 & π‘Ž_2 = βˆ’1, 𝑏_2 = 2, 𝑐_2= 5 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( βˆ’1βˆ’(βˆ’3)&2βˆ’1&5βˆ’5@ βˆ’3&1&5@ βˆ’1&2&5)| = |β– 8(2&1&0@βˆ’3&1&5@βˆ’1&2&5)| = 2[(1Γ—5)βˆ’(2Γ—5)] βˆ’ 1[(βˆ’3Γ—5)βˆ’(βˆ’ 1Γ—5)] + 0 [(βˆ’3Γ—2)βˆ’(βˆ’1Γ—1)] = 2[5βˆ’10]βˆ’ 1[βˆ’ 15βˆ’(βˆ’ 5)] + 0 = 2(βˆ’5) βˆ’1(βˆ’10) = βˆ’10 + 10 = 0 Therefore, the given two lines are coplanar.

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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.