Misc 12 - Find point where line crosses plane 2x + y + z = 7 - Miscellaneous

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  1. Chapter 11 Class 12 Three Dimensional Geometry
  2. Serial order wise
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Misc 12 Find the coordinates of the point where the line through (3, –4, –5) and (2, –3, 1) crosses the plane 2x + y + z = 7. The equation of a line passing through two points A( 𝑥﷮1﷯, 𝑦﷮1﷯, 𝑧﷮1﷯) and B( 𝑥﷮2﷯, 𝑦﷮2﷯, 𝑧﷮2﷯) is 𝒙 − 𝒙﷮𝟏﷯﷮ 𝒙﷮𝟐﷯ − 𝒙﷮𝟏﷯﷯ = 𝒚 − 𝒚﷮𝟏﷯﷮ 𝒚﷮𝟐﷯ − 𝒚﷮𝟏﷯﷯ = 𝒛 − 𝒛﷮𝟏﷯﷮ 𝒛﷮𝟐﷯ − 𝒛﷮𝟏﷯﷯ Given the line passes through the points So, the equation of line is 𝑥 − 3﷮2 − 3﷯ = 𝑦 −(−4)﷮−3 −(−4)﷯ = 𝑧 −(−5)﷮1 −(−5)﷯ 𝒙 − 𝟑﷮−𝟏﷯ = 𝒚 + 𝟒﷮𝟏﷯ = 𝒛 + 𝟓﷮𝟔﷯ = 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 2(−k + 3) + (k − 4) + (6k − 5) = 7 −2k + 6 + k − 4 + 6k − 5 = 7 5k − 3 = 7 5k = 7 + 3 5k = 10 ∴ k = 𝟏𝟎﷮𝟓﷯ = 2 Putting value of k in x, y, z, x = −k + 3 = −2 + 3 = 1 y = k − 4 = 2 − 4 = −2 z = 6k − 5 = 6 × 2 − 5 = 12 − 5 = 7 Therefore, the coordinate of the required point are (1, −2, 7).

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