Last updated at Feb. 1, 2020 by Teachoo
Transcript
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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