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Question
Find the coordinates of the point where the line through (3, – 4, – 5) and (2, –3, 1) crosses the plane passing through three points (2, 2, 1), (3, 0, 1) and (4, –1, 0)
Solution
Equation of plane through three points (2, 2, 1), (3, 0, 1) and (4, –1, 0) is
`[(vecr - (2hati + 2hatj + hatk)]*[(hati - 2hatj) xx (hati - hatj - hatk)]` = 0
i.e. `vecr*(2hati + hatj + hatk)` = 7 or 2x + y + z – 7 = 0 ......(1)
Equation of line through (3, – 4, – 5) and (2, – 3, 1) is
`(x - 3)/(-1) = (y + 4)/1 = (z + 5)/6` ......(2)
Any point on line (2) is `(-lambda + 3, lambda - 4, 6lambda - 5)`.
This point lies on plane (1).
Therefore, `2(-lambda + 3) + (lambda - 4) + (6lambda - 5) - 7` = 0
i.e., `lambda` = z
Hence the required point is (1, – 2, 7).
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