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Find the foot of the perpendicular from the point (1, 2, 0) upon the plane x – 3y + 2z = 9. Hence, find the distance of the point (1, 2, 0) from the given plane.

This question is similar to Question 37 (Choice 2) CBSE Class 12 - Sample Paper 2021 Boards

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Question 13 Find the foot of the perpendicular from the point (1, 2, 0) upon the plane x – 3y + 2z = 9. Hence, find the distance of the point (1, 2, 0) from the given plane. Let point P(x1, y1, z1) be foot of perpendicular from point X (1, 2, 0) Since perpendicular to plane is parallel to normal vector Vector (𝑿𝑷) βƒ— is parallel to normal vector 𝒏 βƒ— Given equation of the plane is x βˆ’ 3y + 2z = 9 So, Normal vector = 𝒏 βƒ— = π’Š Μ‚ βˆ’ 3𝒋 Μ‚ + 2π’Œ Μ‚ Since, (𝑿𝑷) βƒ— and 𝒏 βƒ— are parallel their direction ratios are proportional. Finding direction ratios (𝑿𝑷) βƒ— = (x1 βˆ’ 1)π’Š Μ‚ + (y1 βˆ’ 2)𝒋 Μ‚ + (z1 βˆ’ 0)π’Œ Μ‚ Direction ratios = x1 βˆ’ 1, y1 βˆ’ 2, z1 ∴ a1 = x1 βˆ’ 1 , b1 = y1 βˆ’ 2, c1 = z1 𝒏 βƒ— = 1π’Š Μ‚ βˆ’ 3𝒋 Μ‚ + 2π’Œ Μ‚ Direction ratios = 1, βˆ’3, 2 ∴ a2 = 1 , b2 = βˆ’3, c2 = 2 Since, (𝑿𝑷) βƒ— and 𝒏 βƒ— are parallel their direction ratios are proportional. Finding direction ratios Direction ratios are proportional π‘Ž_1/π‘Ž_2 = 𝑏_1/𝑏_2 = 𝑐_1/𝑐_2 = k (π‘₯_1 βˆ’ 1)/1 = (𝑦_1 βˆ’ 2)/( βˆ’3) = 𝑧_1/2 = k Thus, x1 = k + 1, y1 = βˆ’3k + 2, z1 = 2k Also, point P(x1, y1, z1) lies in the plane. Putting P (k + 1, βˆ’3k + 2, 2k) in equation of plane x βˆ’ 3y + 2z = 9 (k + 1) βˆ’ 3(βˆ’3k + 2) + 2(2k) = 9 k + 1 + 9k βˆ’ 6 + 4k = 9 k + 9k + 4k + 1 βˆ’ 6 = 9 14k βˆ’ 5 = 9 14k = 9 + 5 14k = 14 ∴ k = 1 Thus, x1 = k + 1 = 1 + 1 = 2 y1 = βˆ’3k + 2 = βˆ’3(1) + 2 = βˆ’1 z1 = 2k = βˆ’2(1) = 2 Therefore, coordinate of foot of perpendicular are P (2, βˆ’1, 2) Length of perpendicular X (1, 2, 0) and P (2, βˆ’1, 2) Let of Perpendicular is length of PX PX = √((2βˆ’1)^2+(βˆ’1βˆ’2)^2+(2βˆ’0)^2 ) PX = √(1^2+(βˆ’3)^2+2^2 ) PX = √(1+9+4) PX = βˆšπŸπŸ’ units

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.