Prove that the lines and are coplanar. Also find the equation of the plane containing these lines.

Given : Equations of lines –

Line 1 :

Line 2 :

To Prove : Line 1 & line 2 are coplanar.

To Find : Equation of plane.

Formulae :

1) Coplanarity of two lines :

If two lines are given by,

and

, then these lines are coplanar, if

2) Equation of plane :

The equation of plane containing two coplanar lines

& is given by,

Answer :

Given lines –

Line 1 :

Line 2 :

Here, x₁ = 2 , y₁ = 4 , z₁ = 6 , a₁ = 1 , b₁ = 4 , c₁ = 7

x₂ = -1 , y₂ = -3 , z₂ = -5 , a₂ = 3 , b₂ = 5 , c₂ = 7

Now,

= 21 - 98 + 77

= 0

Hence, given two lines are coplanar.

Equation of plane passing through line 1 and line 2 is given by,

-7x + 14 + 14y - 56 – 7z + 42 = 0

- 7x + 14y – 7z = 0

x – 2y + z = 0

Therefore, equation of plane is

x – 2y + z = 0