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Chapter 11 Class 12 Three Dimensional Geometry

Serial order wise

Last updated at April 16, 2024 by Teachoo

Question 10 Find the vector equation of the plane passing through the intersection of the planes ๐ โ . (๐ ฬ + ๐ ฬ + ๐ ฬ) = 6 and ๐ โ . (2๐ ฬ + 3๐ ฬ + 4๐ ฬ) = โ 5, and the point (1, 1, 1).The vector equation of a plane passing through the intersection of planes ๐ โ. (๐1) โ = d1 and ๐ โ. (๐2) โ = d2 and also through the point (x1, y1, z1) is ๐ โ.((๐๐) โ + ๐(๐๐) โ) = d1 + ๐d2 Given, the plane passes through ๐ โ.(๐ ฬ + ๐ ฬ + ๐ ฬ) = 6 Comparing with ๐ โ.(๐1) โ = d1, (๐๐) โ = ๐ ฬ + ๐ ฬ + ๐ ฬ & d1 = 6 ๐ โ.(2๐ ฬ + 3๐ ฬ + 4๐ ฬ) = โ5 โ๐ โ.(2๐ ฬ + 3๐ ฬ + 4๐ ฬ) = 5 ๐ โ .(โ 2๐ ฬ โ 3๐ ฬ โ 4๐ ฬ) = 5 Comparing with ๐ โ.(๐2) โ = d2 (๐๐) โ = โ 2๐ ฬ โ 3๐ ฬ โ 4๐ ฬ & d2 = 5 Equation of plane is ๐ โ. [(๐ ฬ+๐ ฬ+๐ ฬ )+"๐" (โ2๐ ฬโ3๐ ฬโ4๐ ฬ)] = 6 + ๐5 ๐ โ. [(๐ ฬ" " +๐ ฬ" " +๐ ฬ )โ"๐" (๐๐ ฬ+๐๐ ฬ+๐๐ ฬ)] = 6 + 5๐ Now to find ๐ , put ๐ โ = x๐ ฬ + y๐ ฬ + z๐ ฬ (x๐ ฬ + y๐ ฬ + z๐ ฬ). [(๐ ฬ+๐ ฬ+๐ ฬ )โ"๐" (2๐ ฬ+3๐ ฬ+4๐ ฬ)] = 5๐ + 6 (x๐ ฬ + y๐ ฬ + z๐ ฬ).(๐ ฬ+๐ ฬ+๐ ฬ ) โ ๐ (x๐ ฬ + y๐ ฬ + z๐ ฬ).(2๐ ฬ+3๐ ฬ+4๐ ฬ) = 5๐ + 6 (x ร 1) + (y ร 1) + (z ร 1) โ ๐[(๐ฅร2)+(๐ฆร3)+(๐งร4)] = 5๐ + 6 x + y + z โ ๐[2๐ฅ+3๐ฆ+4๐ง] = 5๐ + 6 x + y + z โ 2๐๐ฅ โ 3๐y โ 4๐z = 5๐ + 6 (1 โ 2๐)x + (1 โ 3๐)y + (1 โ 4๐) z = 5๐ + 6 Since the plane passes through (1, 1, 1), Putting (1, 1, 1) in (2) (1 โ 2๐)x + (1 โ 3๐)y + (1 โ 4๐) z = 5๐ + 6 (1 โ2๐) ร 1 + (1 โ 3๐) ร 1 + (1 โ 4๐) ร 1 = 5๐ + 6 1 โ2๐ + 1 โ 3๐ + 1 โ 4๐= 5๐ + 6 3 โ 9๐ = 5๐ + 6 โ14๐ = 3 โด ๐ = (โ๐)/๐๐ Putting value of ๐ in (1), ๐ โ. [(๐ ฬ" " +" " ๐ ฬ" " +" " ๐ ฬ )โ(( โ3)/14)(2๐ ฬ+3๐ ฬ+"4" ๐ ฬ)]= 6 + 5 ร ( โ3)/14 ๐ โ. [(๐ ฬ+๐ ฬ+" " ๐ ฬ )+3/14(2๐ ฬ+3๐ ฬ+"4" ๐ ฬ)]= 6 โ 15/14 ๐ โ. [๐ ฬ+๐ ฬ" " +๐ ฬ+ 6/14 ๐ ฬ+9/14 ๐ ฬ+12/14 ๐ ฬ ]= 69/14 ๐ โ. [(1+6/14) ๐ ฬ +(1+9/14) ๐ ฬ+(1+12/14) ๐ ฬ ]= 69/14 ๐ โ. [20/14 ๐ ฬ + 23/14 ๐ ฬ + 26/14 ๐ ฬ ]= 69/14 ๐ โ. [1/14(20๐ ฬ+23๐ ฬ+26๐ ฬ)]= 69/14 1/14 ๐ โ. (20๐ ฬ + 23๐ ฬ + 26๐ ฬ) = 69/14 ๐ โ. (20๐ ฬ + 23๐ ฬ + 26๐ ฬ) = 69 Therefore, the vector equation of the required plane is ๐ โ.(๐๐๐ ฬ + ๐๐๐ ฬ + ๐๐๐ ฬ) = ๐๐