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Question

Find the coordinates of the point where the line through the points (3,4,5) and (2,3,1), crosses the plane determined by the points (1,2,3),(4,2,3) and (0,4,3).

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Solution

The line through (3,4,5) and (2,3,1) is given by

x323=y+43+4=z+51+5

Or x31=y+41=z+56 ...(i)

The plane determined by the points (1,2,3),(4,2,3) and (0,4,3) is:

∣ ∣x1y2z3412233014233∣ ∣=0

∣ ∣x1y2z3306120∣ ∣=0

(x1)(0+12)(y2)(06)+(z3)(60)=0

12(x1)+6(y2)+6(z3)=0

12x+6y+6z42=0

or 2x+y+z7=0 ...(ii)

P(μ+3,μ4,6μ5) is the general point for line (i)

If this point lies on plane (ii), we have

2μ+6+μ4+6μ57=0μ=2

P(1,2,7) is the point of intersection.


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