Answer :

Given: - Two lines equation: and


To find: - Intersection point


We have,



x = λ, y = 2λ + 2 and z = 3λ – 3


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


(λ, 2λ + 2, 3λ – 3)


The equation of the 2nd line is



x = 2μ + 2, y = 3μ + 6 and z = 4μ + 3


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


(2μ + 2, 3μ + 6, 4μ + 3)


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


Therefore for some value of λ and μ, we have


λ = 2μ + 2 , 2λ + 2 = 3μ + 6, and 3λ – 3 = 4μ + 3


λ = 2μ + 2 ……(i)


2λ – 3μ = 4 ……(ii)


and 3λ – 4μ = 6 ……(iii)


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


2(2μ + 2) – 3μ = 4


4μ + 4 – 3μ = 4


μ = 0


Now putting value of μ in eq i, we get


λ = 2μ + 2


λ = 2(0) + 2


λ = 2


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


Check


3λ – 4μ = 6


3(2) = 6 ;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


λ, 2λ + 2, 3λ – 3


2, 6, 3


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