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Question

Prove that the lines x21=y44=z67 and x+13=y+35=z+57 are coplanar. Also, find the equation of the plane containing these two lines.

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Solution


If two line are coplanar then
∣ ∣x2x1y2y1z2z111m1n1l2m2n2∣ ∣=0
Here
(x1,y1,z1)=(2,4,6)
(x2,y2,z2)=(1,3,5)
(l1,m1,n1)=(1,4,7)
(l2,m2,n2)=(3,5,7)
Substituting the values
=∣ ∣3711147357∣ ∣
=3(2835)+7(721)11(512)
=2198+77
=9898
=0
Hence the lines are coplanar.

The equation of the plane containing these lines is
∣ ∣x2y4z6147357∣ ∣=0
(x2)(2835)(y4)(721)+(z6)(512)=0
(x2)(7)(y4)(14)+(z6)(7)=0
7x+14+14y567z+42=0
7x+14y7z=0
x2y+z=0
Hence this is the equation of the plane.

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