Q. 4

Find the values of λ and μ so that the points A(3, 2, -4), B(9, 8, -10) and C(λ, μ -6) are collinear.

Answer :

Given -


A = (3,2,-4)


B = (9,8,-10)


C = (λ,μ,-6)


To find – The value of λ and μ so that A, B and C are collinear


Formula to be used – If P = (a,b,c) and Q = (a’,b’,c’),then the direction ratios of the line PQ is given by ((a’-a),(b’-b),(c’-c))


The direction ratios of the line AB can be given by


((9-3),(8-2),(-10+4))


=(6,6,-6)


Similarly, the direction ratios of the line BC can be given by


((λ-9),(μ-8),(-6+10))


=(λ-9,μ-8,4)


Tip – If it is shown that direction ratios of AB=α times that of BC , where λ is any arbitrary constant, then the condition is sufficient to conclude that points A, B and C will be collinear.


So, d.r. of AB


=(6,6,-6)


=(-6/4)Χ(-4,-4,4)


=(-3/2)Хd.r. of BC


Since, A, B and C are collinear,





And,





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