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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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Transcript

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 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.