Given: - Two lines equation: and

To find: - Intersection point

We have,

⇒ x = 3λ + 1, y = 2λ – 1 and z = 5λ + 1

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

(3λ + 1, 2λ – 1, 5λ + 1)

The equation of the 2^nd line is

⇒ x = 4μ – 2, y = 3μ + 1 and z = – 2μ – 1

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

(4μ – 2, 3μ + 1, – 2μ – 1)

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

Therefore for some value of λ and μ, we have

⇒ 3λ + 1 = 4μ – 2 , 2λ – 1 = 3μ + 1, and 5λ + 1 = – 2μ – 1

⇒ ……(i)

⇒ 2λ – 3μ = 2 ……(ii)

and 5λ + 2μ = – 2 ……(iii)

putting value of λ from eq i in eq ii, we get

⇒ – μ = 12

⇒ μ = – 12

Now putting value of μ in eq i, we get

⇒

⇒

⇒ λ = – 17

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

Check

LHS

= 5λ + 2μ

= 5( – 17) + 2( – 12)

= – 85 – 24

= – 109

≠ RHS

Hence intersection point does not exist or line do not intersects