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Misc 14 - If points (1, 1, p) and (-3, 0, 1) be equidistant from plane

Misc 14 - Chapter 11 Class 12 Three Dimensional Geometry - Part 2
Misc 14 - Chapter 11 Class 12 Three Dimensional Geometry - Part 3
Misc 14 - Chapter 11 Class 12 Three Dimensional Geometry - Part 4


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Misc 14 If the points (1, 1 , p) and (ā€“3 , 0, 1) be equidistant from the plane š‘Ÿ āƒ—. (3š‘– Ģ‚ + 4š‘— Ģ‚ āˆ’ 12š‘˜ Ģ‚) + 13 = 0, then find the value of p. The distance of a point with position vector š‘Ž āƒ— from the plane š‘Ÿ āƒ—.š‘› āƒ— = d is |(š’‚ āƒ—.š’ āƒ— āˆ’ š’…)/|š’ āƒ— | | Given, the points are The equation of plane is š‘Ÿ āƒ—. (3š‘– Ģ‚ + 4š‘— Ģ‚ āˆ’ 12š‘˜ Ģ‚) + 13 = 0 š‘Ÿ āƒ—.(3š‘– Ģ‚ + 4š‘— Ģ‚ āˆ’ 12š‘˜ Ģ‚) = āˆ’13 (1, 1, p) So, (š‘Ž_1 ) āƒ— = 1š‘– Ģ‚ + 1š‘— Ģ‚ + pš‘˜ Ģ‚ (āˆ’3, 0, 1) So, (š‘Ž_2 ) āƒ— = āˆ’3š‘– Ģ‚ + 0š‘— Ģ‚ + 1š‘˜ Ģ‚ ā€“š‘Ÿ āƒ—.(3š‘– Ģ‚ + 4š‘— Ģ‚ āˆ’ 12š‘˜ Ģ‚) = 13 š’“ āƒ—.(ā€“3š’Š Ģ‚ ā€“ 4š’‹ Ģ‚ + 12š’Œ Ģ‚) = 13 Comparing with š‘Ÿ āƒ—.š‘› āƒ— = d, š‘› āƒ— = āˆ’3š‘– Ģ‚ āˆ’ 4š‘— Ģ‚ + 12š‘˜ Ģ‚ d = 13 Magnitude of š‘› āƒ— = āˆš((āˆ’3)^2+(āˆ’4)^2+怖12怗^2 ) |š‘› āƒ— | = āˆš(9+16+144) = āˆš169 = 13 Distance of point (š’‚šŸ) āƒ— from plane |((š‘Ž1) āƒ—"." š‘› āƒ—" " āˆ’ š‘‘)/|š‘› āƒ— | | = |((1š‘– Ģ‚ + 1š‘— Ģ‚ + š‘š‘˜ Ģ‚ ).(āˆ’3š‘– Ģ‚āˆ’4š‘— Ģ‚+12š‘˜ Ģ‚ )āˆ’13)/13| = |((1Ɨāˆ’3)+(1Ɨāˆ’4) +(š‘Ć—12)āˆ’13)/13| = |(āˆ’3āˆ’4+12š‘āˆ’13)/13| = |(12š‘ āˆ’ 20)/13| Distance of point (š’‚šŸ) āƒ— from plane |((š‘Ž2) āƒ—"." š‘› āƒ— āˆ’ š‘‘)/|š‘› āƒ— | | = |((āˆ’3š‘– Ģ‚ +0š‘— Ģ‚ +1š‘˜ Ģ‚ ).(āˆ’3š‘– Ģ‚āˆ’4š‘— Ģ‚+12š‘˜ Ģ‚ )āˆ’13)/13| = |((āˆ’3Ɨāˆ’3)+(0Ɨāˆ’4) +(1Ɨ12)āˆ’13)/13| = |(9 + 0 +12āˆ’13)/13| = |8/13| = 8/13 Since the plane is equidistance from both the points, |(šŸšŸš’‘ āˆ’ šŸšŸŽ)/šŸšŸ‘| = šŸ–/šŸšŸ‘ |12š‘āˆ’20| = 8 (12p ā€“ 20) = Ā± 8 12p āˆ’ 20 = 8 12p = 8 + 20 12p = 28 p = 28/12 p = 7/3 12p āˆ’ 20 = āˆ’8 12p = āˆ’8 + 20 12p = 12 p = 12/12 p = 1 Answer does not match at end. If mistake, please comment

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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.