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Example 28 - Find distance between point P(6, 5, 9) and plane

Example  28 - Chapter 11 Class 12 Three Dimensional Geometry - Part 2
Example  28 - Chapter 11 Class 12 Three Dimensional Geometry - Part 3 Example  28 - Chapter 11 Class 12 Three Dimensional Geometry - Part 4


Transcript

Example 28 Find the distance between the point P(6, 5, 9) and the plane determined by the points A (3, – 1, 2), B (5, 2, 4) and C(– 1, – 1, 6).The equation of a plane passing through points A(π‘₯_1, 𝑦_1, 𝑧_1) B(π‘₯_2, 𝑦_2, 𝑧_2) and C (π‘₯_3, 𝑦_3, 𝑧_3) is |β– 8(π’™βˆ’π’™_𝟏&π’šβˆ’π’š_𝟏&π’›βˆ’π’›_𝟏@𝒙_πŸβˆ’π’™_𝟏&π’š_πŸβˆ’π’š_𝟏&𝒛_πŸβˆ’π’›_𝟏@𝒙_πŸ‘βˆ’π’™_𝟏&π’š_πŸ‘βˆ’π’š_𝟏&𝒛_πŸ‘βˆ’π’›_𝟏 )| = 0 Given, the three points are A(3, βˆ’1, 2) π‘₯_1= 3, 𝑦_1= βˆ’1, 𝑧_1= 2 B(5, 2, 4) π‘₯_2= 5, 𝑦_2 = 2, 𝑧_2= 4 C(–1, –1, 6) π‘₯_3= βˆ’1, 𝑦_3= βˆ’1, 𝑧_3= 6 Equation of plane is |β– 8(π‘₯βˆ’3&π‘¦βˆ’(βˆ’1)&π‘§βˆ’[email protected]βˆ’3&2βˆ’(βˆ’1)&4βˆ’[email protected]βˆ’1βˆ’3&βˆ’1βˆ’(βˆ’1)&6βˆ’2)| = 0 |β– 8(π‘₯βˆ’3&𝑦+1&π‘§βˆ’[email protected]&3&[email protected]βˆ’4&0&4)| = 0 (x βˆ’ 3)[(3Γ—4)βˆ’(0Γ—2)] βˆ’ (y + 1) [(2Γ—4)βˆ’(βˆ’4Γ—2)] + (z βˆ’ 2) [(2Γ—0)βˆ’(βˆ’4Γ—3)] (x βˆ’ 3)[12βˆ’0] βˆ’ (y + 1) [8+8] +(π‘§βˆ’2) [0+12] = 0 12(x βˆ’ 3) – 16(y + 1) + 12(z βˆ’ 2) = 0 3(x βˆ’ 3) – 4(y + 1) + 3(z βˆ’ 2) = 0 3x βˆ’ 9 – 4y – 4 + 3z βˆ’ 6 = 0 3x – 4y + 3z βˆ’ 19 = 0 Therefore, equation of plane is 3x – 4y + 3z = 19 Now, the distance between a point P(π‘₯_1, 𝑦_1, 𝑧_1) and the plane Ax + By + Cz = D is |(𝑨𝒙_𝟏 + π‘©π’š_𝟏 + π‘ͺ𝒛_πŸβˆ’ 𝑫)/√(𝑨^𝟐 + 𝑩^𝟐 + π‘ͺ^𝟐 )| Given, the point is P(6, 5, 9) So, π‘₯_1= 6 , 𝑦_1= 5, 𝑧_1= 9 The equation of plane is 3x – 4y + 3z = 19 Comparing with Ax + By + Cz = D, A = 3, B = –4, C = 3, D = 19 Now, Distance of the point from the plane = |((3 Γ— 6) + (βˆ’4 Γ— 5) + (3 Γ— 9) βˆ’ 19)/√(3^2 + 4^2 + 3^2 )| =|(18 βˆ’ 20 + 27 βˆ’ 19)/√(9 + 16 + 9)| =|6/√34| = 6/√34 = 6/√34 Γ— √34/√34 = (6√34)/34 = (πŸ‘βˆšπŸ‘πŸ’)/πŸπŸ•

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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.