Slide41.JPG

Slide42.JPG
Slide43.JPG Slide44.JPG

  1. Chapter 11 Class 12 Three Dimensional Geometry
  2. Serial order wise

Transcript

Example 28 Find the distance between the point P(6, 5, 9) and the plane determined by the points A (3, โ€“ 1, 2), B (5, 2, 4) and C(โ€“ 1, โ€“ 1, 6).The equation of a plane passing through points A(๐‘ฅ_1, ๐‘ฆ_1, ๐‘ง_1) B(๐‘ฅ_2, ๐‘ฆ_2, ๐‘ง_2) and C (๐‘ฅ_3, ๐‘ฆ_3, ๐‘ง_3) is |โ– 8(๐’™โˆ’๐’™_๐Ÿ&๐’šโˆ’๐’š_๐Ÿ&๐’›โˆ’๐’›_๐Ÿ@๐’™_๐Ÿโˆ’๐’™_๐Ÿ&๐’š_๐Ÿโˆ’๐’š_๐Ÿ&๐’›_๐Ÿโˆ’๐’›_๐Ÿ@๐’™_๐Ÿ‘โˆ’๐’™_๐Ÿ&๐’š_๐Ÿ‘โˆ’๐’š_๐Ÿ&๐’›_๐Ÿ‘โˆ’๐’›_๐Ÿ )| = 0 Given, the three points are A(3, โˆ’1, 2) ๐‘ฅ_1= 3, ๐‘ฆ_1= โˆ’1, ๐‘ง_1= 2 B(5, 2, 4) ๐‘ฅ_2= 5, ๐‘ฆ_2 = 2, ๐‘ง_2= 4 C(โ€“1, โ€“1, 6) ๐‘ฅ_3= โˆ’1, ๐‘ฆ_3= โˆ’1, ๐‘ง_3= 6 Equation of plane is |โ– 8(๐‘ฅโˆ’3&๐‘ฆโˆ’(โˆ’1)&๐‘งโˆ’2@5โˆ’3&2โˆ’(โˆ’1)&4โˆ’2@โˆ’1โˆ’3&โˆ’1โˆ’(โˆ’1)&6โˆ’2)| = 0 |โ– 8(๐‘ฅโˆ’3&๐‘ฆ+1&๐‘งโˆ’2@2&3&2@โˆ’4&0&4)| = 0 (x โˆ’ 3)[(3ร—4)โˆ’(0ร—2)] โˆ’ (y + 1) [(2ร—4)โˆ’(โˆ’4ร—2)] + (z โˆ’ 2) [(2ร—0)โˆ’(โˆ’4ร—3)] (x โˆ’ 3)[12โˆ’0] โˆ’ (y + 1) [8+8] +(๐‘งโˆ’2) [0+12] = 0 12(x โˆ’ 3) โ€“ 16(y + 1) + 12(z โˆ’ 2) = 0 3(x โˆ’ 3) โ€“ 4(y + 1) + 3(z โˆ’ 2) = 0 3x โˆ’ 9 โ€“ 4y โ€“ 4 + 3z โˆ’ 6 = 0 3x โ€“ 4y + 3z โˆ’ 19 = 0 Therefore, equation of plane is 3x โ€“ 4y + 3z = 19 Now, the distance between a point P(๐‘ฅ_1, ๐‘ฆ_1, ๐‘ง_1) and the plane Ax + By + Cz = D is |(๐‘จ๐’™_๐Ÿ + ๐‘ฉ๐’š_๐Ÿ + ๐‘ช๐’›_๐Ÿโˆ’ ๐‘ซ)/โˆš(๐‘จ^๐Ÿ + ๐‘ฉ^๐Ÿ + ๐‘ช^๐Ÿ )| Given, the point is P(6, 5, 9) So, ๐‘ฅ_1= 6 , ๐‘ฆ_1= 5, ๐‘ง_1= 9 The equation of plane is 3x โ€“ 4y + 3z = 19 Comparing with Ax + By + Cz = D, A = 3, B = โ€“4, C = 3, D = 19 Now, Distance of the point from the plane = |((3 ร— 6) + (โˆ’4 ร— 5) + (3 ร— 9) โˆ’ 19)/โˆš(3^2 + 4^2 + 3^2 )| =|(18 โˆ’ 20 + 27 โˆ’ 19)/โˆš(9 + 16 + 9)| =|6/โˆš34| = 6/โˆš34 = 6/โˆš34 ร— โˆš34/โˆš34 = (6โˆš34)/34 = (๐Ÿ‘โˆš๐Ÿ‘๐Ÿ’)/๐Ÿ๐Ÿ•

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.