Misc 17 - Plane which contains line of intersection of planes - Miscellaneous

Slide64.JPG
Slide65.JPG Slide66.JPG Slide67.JPG

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

Transcript

Misc 17 Find the equation of the plane which contains the line of intersection of the planes 𝑟﷯ . ( 𝑖﷯ + 2 𝑗﷯ + 3 𝑘﷯) – 4 = 0 , 𝑟﷯ . (2 𝑖﷯ + 𝑗﷯ – 𝑘﷯) + 5 = 0 and which is perpendicular to the plane 𝑟﷯ . (5 𝑖﷯ + 3 𝑗﷯ – 6 𝑘﷯) + 8 = 0 . Equation of a plane passing through the intersection of the places A1x + B1y + C1z = d1 and A2x + B2y + C2z = d2 is (A1x + B1y + C1z − d1) + 𝜆 (A2x + B2y + C2z – d2) = 0 Converting equation of planes to Cartesian form to find A1, B1, C1, d1 & A2, B2, C2, d2 Equation of plane is (A1x + B1y + C1z − d1) + 𝜆 (A2x + B2y + C2z = d2) = 0 Putting values (1x + 2y + 3z − 4) + 𝜆 ( − 2x − 1y + 1z − 5) = 0 (1 − 2𝜆) x + (2 − 𝜆)y + (3 + 𝜆) z + ( −4 − 5𝜆) = 0 Now, the plane is perpendicular to the plane 𝑟﷯.(5 𝑖﷯ + 3 𝑗﷯ − 6 𝑘﷯) + 8 = 0 So, normal to plane 𝑁﷯ will be perpendicular to normal 𝑛﷯ of 𝑟﷯.(5 𝑖﷯ + 3 𝑗﷯ − 6 𝑘﷯) + 8 = 0 Now, 𝑟﷯.(5 𝑖﷯ + 3 𝑗﷯ − 6 𝑘﷯) + 8 = 0 𝑟﷯ .(5 𝑖﷯ + 3 𝑗﷯ − 6 𝑘﷯) = –8 − 𝑟﷯ .(5 𝑖﷯ + 3 𝑗﷯ − 6 𝑘﷯) = 8 𝑟﷯ .( −5 𝑖﷯ − 3 𝑗﷯ + 6 𝑘﷯) = 8 Finding direction cosines of 𝑁﷯ & 𝑛﷯ Since, 𝑁﷯ is perpendicular to 𝑛﷯ 𝑎1 𝑎2 + b1 b2 + c1 c2 = 0 (1 − 2𝜆) × −5 + (2 − 𝜆) × −3 + (3 + 𝜆) × 6 = 0 − 5 + 10𝜆 − 6 + 3𝜆 + 18 + 6𝜆 = 0 19𝜆 + 7 = 0 ∴ 𝜆 = −𝟕﷮𝟏𝟗﷯ Putting value of 𝜆 in (1), (1 − 2𝜆) x + (2 − 𝜆)y + (3 + 𝜆) z + ( −4 − 5𝜆) = 0 1−2 × −7﷮19﷯﷯ x + 2− −7﷮19﷯﷯﷯y + 3+ − 7﷮19﷯﷯﷯z + −4−5× −7﷮19﷯﷯ = 0 1 + 14﷮19﷯﷯ x + 2 + 7﷮19﷯﷯y + 3 − 7﷮19﷯﷯z + − 4 + 35﷮19﷯﷯ = 0 33﷮19﷯ x + 45﷮19﷯ y + 50﷮19﷯ z − 41﷮19﷯ = 0 1﷮19﷯ (33x + 45y + 50z − 41) = 0 33x + 45y + 50z − 41 = 0 Therefore, the equation of the plane is 33x + 45y + 50z = 41.

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 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.
Jail