Question 27 (OR 2 nd question)

Show that the line of intersection of the planes x + 2y + 3z = 8 and 2x + 3y + 4z = 11 is coplanar with the line (x + 1)/1 = (y + 1)/2 = (z + 1)/3 . Also find the equation of the plane containing them.

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Question 27 (OR 2nd question) Show that the line of intersection of the planes x + 2y + 3z = 8 and 2x + 3y + 4z = 11 is coplanar with the line (๐‘ฅ + 1)/1 = (๐‘ฆ + 1)/2 = (๐‘ง + 1)/3 . Also find the equation of the plane containing them. Given planes x + 2y + 3z = 8 2x + 3y + 4z = 11 and line (๐‘ฅ + 1)/1 = (๐‘ฆ + 1)/2 = (๐‘ง + 1)/3 Let the plane passing through intersection of planes (1) and (2) be (x + 2y + 3z โ€“ 8) + k(2x + 3y + 4z โ€“ 11) = 0 Now, we have to show line of intersection of the planes (1) and (2) is coplanar with line (3) Line of intersection of planes will lie in plane (4) If line (3) is coplanar with line of intersection of planes, Then, Line (3)โ€™s point (โ€“1, โ€“1, โ€“1) passes through plane (4) and Normal of Plane (4) is perpendicular to parallel vector of line (3) Putting (โ€“1, โ€“1, โ€“1) in equation of plane from (4) (x + 2y + 3z โ€“ 8) + k(2x + 3y + 4z โ€“ 11) = 0 Putting x = โ€“1, y = โ€“1, z = โ€“1 ((โ€“1) + 2(โ€“1) + 3(โ€“1) โ€“ 8) + k(2(โ€“1) + 3(โ€“1) + 4(โ€“1) โ€“ 11) = 0 (โ€“1 โ€“ 2 โ€“ 3 โ€“ 8) + k(โ€“2 โ€“ 3 โ€“ 4 โ€“ 11) = 0 โ€“14 โ€“ 20k = 0 โ€“20k = 14 k =14/(โˆ’20) k = (โˆ’7)/10 Putting k = (โˆ’7)/10 in (4) (x + 2y + 3z โ€“ 8) + ((โˆ’7)/10)(2x + 3y + 4z โ€“ 11) = 0 10(x + 2y + 3z โ€“ 8) โ€“ 7(2x + 3y + 4z โ€“ 11) = 0 10x + 20y + 30z โ€“ 80 โ€“ 14x โ€“ 21y โ€“ 28z + 77 = 0 โ€“4x โ€“ y + 2z โ€“ 3 = 0 โ€“4x โ€“ y + 2z โ€“3 = 0 4x + y โ€“ 2z + 3 = 0 Thus, equation of plane is 4x + y โ€“ 2z + 3 = 0 Now, we check if line (3) (๐‘ฅ + 1)/1 = (๐‘ฆ + 1)/2 = (๐‘ง + 1)/3 is perpendicular to plane (5) Direction ratios of normal of plane = 4, 1, โ€“2 So, a1 = 4, b1 = 1, c1 = โ€“2 Direction ratios of line = 1, 2, 3 So, a2 = 1, b2 = 2, c2 = 3 Finding a1a2 + b1b2 + c1c2 = 4 ร— 1 + 1 ร— 2 + (โ€“2) ร— 3 = 4 + 2 โ€“ 6 = 0 Since a1a2 + b1b2 + c1c2 = 0 Normal to the plane and line is perpendicular Hence line (3) lies in the plane passing through intersection of planes Therefore, the two lines are coplanar and the equation of the plane containing them is 4x + y โ€“ 2z + 3 = 0

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.