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Chapter 11 Class 12 Three Dimensional Geometry

Serial order wise

Last updated at Feb. 1, 2020 by Teachoo

Example 20 Find the vector equation of the plane passing through the intersection of the planes π β . (π Μ + π Μ + π Μ) = 6 and π β . (2π Μ + 3π Μ + 4π Μ) = β 5, and the point (1, 1, 1).The vector equation of a plane passing through the intersection of planes π β. (π1) β = d1 and π β. (π2) β = d2 and also through the point (x1, y1, z1) is π β.((ππ) β + π(ππ) β) = d1 + πd2 Given, the plane passes through π β.(π Μ + π Μ + π Μ) = 6 Comparing with π β.(π1) β = d1, (ππ) β = π Μ + π Μ + π Μ & d1 = 6 π β.(2π Μ + 3π Μ + 4π Μ) = β5 βπ β.(2π Μ + 3π Μ + 4π Μ) = 5 π β .(β 2π Μ β 3π Μ β 4π Μ) = 5 Comparing with π β.(π2) β = d2 (ππ) β = β 2π Μ β 3π Μ β 4π Μ & d2 = 5 Equation of plane is π β. [(π Μ+π Μ+π Μ )+"π" (β2π Μβ3π Μβ4π Μ)] = 6 + π5 π β. [(π Μ" " +π Μ" " +π Μ )β"π" (ππ Μ+ππ Μ+ππ Μ)] = 6 + 5π Now to find π , put π β = xπ Μ + yπ Μ + zπ Μ (xπ Μ + yπ Μ + zπ Μ). [(π Μ+π Μ+π Μ )β"π" (2π Μ+3π Μ+4π Μ)] = 5π + 6 (xπ Μ + yπ Μ + zπ Μ).(π Μ+π Μ+π Μ ) β π (xπ Μ + yπ Μ + zπ Μ).(2π Μ+3π Μ+4π Μ) = 5π + 6 (x Γ 1) + (y Γ 1) + (z Γ 1) β π[(π₯Γ2)+(π¦Γ3)+(π§Γ4)] = 5π + 6 x + y + z β π[2π₯+3π¦+4π§] = 5π + 6 x + y + z β 2ππ₯ β 3πy β 4πz = 5π + 6 (1 β 2π)x + (1 β 3π)y + (1 β 4π) z = 5π + 6 Since the plane passes through (1, 1, 1), Putting (1, 1, 1) in (2) (1 β 2π)x + (1 β 3π)y + (1 β 4π) z = 5π + 6 (1 β2π) Γ 1 + (1 β 3π) Γ 1 + (1 β 4π) Γ 1 = 5π + 6 1 β2π + 1 β 3π + 1 β 4π= 5π + 6 3 β 9π = 5π + 6 β14π = 3 β΄ π = (βπ)/ππ Putting value of π in (1), π β. [(π Μ" " +" " π Μ" " +" " π Μ )β(( β3)/14)(2π Μ+3π Μ+"4" π Μ)]= 6 + 5 Γ ( β3)/14 π β. [(π Μ+π Μ+" " π Μ )+3/14(2π Μ+3π Μ+"4" π Μ)]= 6 β 15/14 π β. [π Μ+π Μ" " +π Μ+ 6/14 π Μ+9/14 π Μ+12/14 π Μ ]= 69/14 π β. [(1+6/14) π Μ +(1+9/14) π Μ+(1+12/14) π Μ ]= 69/14 π β. [20/14 π Μ + 23/14 π Μ + 26/14 π Μ ]= 69/14 π β. [1/14(20π Μ+23π Μ+26π Μ)]= 69/14 1/14 π β. (20π Μ + 23π Μ + 26π Μ) = 69/14 π β. (20π Μ + 23π Μ + 26π Μ) = 69 Therefore, the vector equation of the required plane is π β.(πππ Μ + πππ Μ + πππ Μ) = ππ