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Last updated at Feb. 1, 2020 by Teachoo
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Misc 12 Find the coordinates of the point where the line through (3, โ4, โ5) and (2, โ3, 1) crosses the plane 2x + y + z = 7. The equation of a line passing through two points A(๐ฅ_1, ๐ฆ_1, ๐ง_1) and B(๐ฅ_2, ๐ฆ_2, ๐ง_2) is (๐ โ ๐_๐)/(๐_๐ โ ๐_๐ ) = (๐ โ ๐_๐)/(๐_๐ โ ๐_๐ ) = (๐ โ ๐_๐)/(๐_๐ โ ๐_๐ ) Given the line passes through the points A (3, โ4, โ5) โด๐ฅ_1 = 3, ๐ฆ_1= โ4, ๐ง_1= โ5 B (2, โ3, 1) โด๐ฅ_2 = 2, ๐ฆ_2= โ3, ๐ง_2= 1 So, the equation of line is (๐ฅ โ 3)/(2 โ 3) = (๐ฆ โ (โ4))/(โ3 โ (โ4)) = (๐ง โ (โ5))/(1 โ (โ5)) (๐ โ ๐)/(โ๐) = (๐ + ๐)/๐ = (๐ + ๐)/๐ = k So, Let (x, y, z) be the coordinates of the point where the line crosses the plane 2x + y + z = 7 Putting value of x, y, z, from (1) in the equation of plane, 2x + y + z = 7 x = โk + 3 2(โk + 3) + (k โ 4) + (6k โ 5) = 7 โ2k + 6 + k โ 4 + 6k โ 5 = 7 5k โ 3 = 7 5k = 7 + 3 5k = 10 โด k = ๐๐/๐ = 2 Putting value of k in x, y, z, x = โk + 3 = โ2 + 3 = 1 y = k โ 4 = 2 โ 4 = โ2 z = 6k โ 5 = 6 ร 2 โ 5 = 12 โ 5 = 7 Therefore, the coordinate of the required point are (1, โ2, 7).
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