Q. 185.0( 1 Vote )

# Show that the lines and intersect. Find their point of intersection.

Answer :

Given:

and

Now, let

⇒ x – 1 = 3λ, y – 1 = -λ and z + 1 = 0

⇒ x = 3λ + 1, y = -λ + 1 and z = -1

Let

⇒ x – 4 = 2μ, y = 0 and z + 1 = 3μ

⇒ x = 2μ + 4, y = 0 and z = 3μ – 1

If lines intersect, then they have a common point for some value of λ and μ

So, 3λ + 1 = 2μ + 4 [comparing the values of x]

⇒ 3λ – 2μ = 4 – 1

⇒ 3λ – 2μ = 3 …(i)

and –λ + 1 = 0 [comparing the values of y]

⇒ λ = 1

and -1 = 3μ – 1

⇒ -1 + 1 = 3μ

⇒ μ = 0

Now, putting the values of λ and μ in eq. (i), we get

3(1) – 2(0) = 3

⇒ 3 = 3

Since, λ = 1 and μ = 0 satisfy equation (i) so the given lines intersect.

Now, the point of intersection are

x = 3λ + 1, y = -λ + 1 and z = -1

⇒ x = 3(1) + 1, y = -(1) + 1 and z = -1

⇒ x = 4 , y = 0 and z = -1

Point of intersection are (4, 0, -1)

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