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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’2@5βˆ’3&2βˆ’(βˆ’1)&4βˆ’2@βˆ’1βˆ’3&βˆ’1βˆ’(βˆ’1)&6βˆ’2)| = 0 |β– 8(π‘₯βˆ’3&𝑦+1&π‘§βˆ’2@2&3&2@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 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.