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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 the points (2,2,1),(3,0,1) and (4,1,0).

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Solution

Equation of line is: x31=y+41=z+56=λ
General point x=λ+3,y=λ+4,z=6λ5
Equation of plane: ∣ ∣x2y2z1320211421201∣ ∣=0
∣ ∣x2y2z1120231∣ ∣=0
=(x2)(20)(y2)(10)+(z1)(3+4)=0
=2x4+y2+z1=0
=2x+y+z7=0---- (1)
General point of line is a point on (1),

Therefore, 2(λ+3)+(λ4)+(6λ5)7=0

=2λ+6+λ4+6λ57=0

=5λ10=0

=λ=2
Point of intersection of line and plane (1,2,7).

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