Given: - Two lines equation: and ; z = 3

We have,

⇒ x = 2λ + 1, y = 3λ – 1 and z = λ

So, the coordinates of a general point on this line are

(2λ + 1, 3λ – 1, λ)

The equation of the 2^nd line is

⇒ x = 5μ + 1, y = μ + 2 and z = 3

So, the coordinates of a general point on this line are

(5μ + 1, μ + 2, 3)

If the lines intersect, then they must have a common point.

Therefore for some value of λ and μ, we have

⇒ 2λ + 1 = 5μ + 1 , 3λ – 1 = μ + 2, and λ = 3

⇒ 2λ – 5μ = 0 ……(i)

⇒ 3λ – μ = 3 ……(ii)

and λ = 3 ……(iii)

putting the value of λ from eq iii in eq ii, we get

⇒ 3λ – μ = 3

⇒ 3(3) – μ = 3

⇒ μ = 6

As we can see by putting the value of λ and μ in eq i, that it does not satisfy the equation.

Check

⇒ 2λ – 5μ = 0

⇒ 2(3) – 5(6) = 0

⇒ – 24 = 0

⇒ LHS≠RHS; Hence intersection point exists or line does not intersects.