Determine whether the following pair of lines intersect or not :

and .

Given: - Two lines equation: and

To find: - Intersection point

We have,

⇒ x = 4λ + 5, y = 4λ + 7 and z = – 5λ – 3

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

(4λ + 5, 4λ + 7, – 5λ – 3)

The equation of the 2^nd line is

⇒ x = 7μ + 8, y = μ + 4 and z = 3μ + 5

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

(7μ + 8, μ + 4, 3μ + 5)

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

Therefore for some value of λ and μ, we have

⇒ 4λ + 5 = 7μ + 8 , 4λ + 7 = μ + 4, and – 5λ – 3 = 3μ + 5

⇒ 4λ – 7μ = 3 ……(i)

⇒ μ = 4λ + 3 ……(ii)

and – 5λ – 3μ = 8 ……(iii)

putting the value of μ from eq ii in eq i, we get

⇒ 4λ – 7μ = 3

⇒ 4λ – 7(4λ + 3) = 3

⇒ 4λ – 28λ – 21 = 3

⇒ – 24λ = 24

⇒ λ = – 1

Now putting the value of λ in eq ii, we get

⇒ μ = 4λ + 3

⇒ μ = 4( – 1) + 3

⇒ μ = – 1

As we can see by putting the value of λ and μ in eq iii, that it satisfy the equation.

Check

⇒ – 5λ – 3μ = 8

⇒ – 5( – 1) – 3( – 1) = 8

⇒ 5 + 3 = 8

⇒ 8 = 8

⇒ LHS = RHS ;Hence intersection point exist or line do intersects

We can find intersecting point by putting values of μ or λ in any one general point equation

Thus,

Intersection point

4λ + 5, 4λ + 7, – 5λ – 3

4( – 1) + 5, 4( – 1) + 7, – 5( – 1) – 3

1, 3, 2