Answer :

Given: - Line joining A(0, – 1, – 1) and B(4, 5, 1).


Line joining C(3, 9, 4) and D( – 4, 4, 4).


To Prove: - Both lines intersects


Proof: - Equation of a line joined by two points A(x1,y1,z1) and B(x2,y2,z2) is given by



Now equation of line joining A(0, – 1, – 1) and B(4, 5, 1)



=


x = 4λ, y = 6λ – 1 and z = 2λ – 1


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


(4λ, 6λ – 1, 2λ – 1)


And equation of line joining C(3, 9, 4) and D( – 4, 4, 4)



=


x = – 7μ + 3, y = – 5μ + 9 and z = 4


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


( – 7μ + 3, – 5μ + 9, 4)


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


Therefore for some value of λ and μ, we have


4λ = – 7μ + 3 , 6λ – 1 = – 5μ + 9, and 2λ – 1 = 4


……(i)


6λ + 5μ = 10 ……(ii)


and 2λ = 5 ……(iii)


from eq iii, we get



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



– 1


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


Check


6λ + 5μ = 10




10 = 10


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


– 7μ + 3, – 5μ + 9, 4


– 7( – 1) + 3, – 5( – 1) + 9, 4


10, 14, 4


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