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Question

Find the value of λ such that points P(1,2),Q(2,3) and R(λ+1,λ) are not forming a triangle?

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Solution

Pi+2j
Q2i+3j
R(λ+1)i+λj
¯¯¯¯¯¯¯¯PQ=(21)i+(32)j
=3i+j
¯¯¯¯¯¯¯¯¯QR=(λ+1+2)i+(λ3)j
=(λ+3)i+(λ3)j
If PQR form a triangle,
¯¯¯¯¯¯¯¯PQ+¯¯¯¯¯¯¯¯¯QR=¯¯¯¯¯¯¯¯PR
¯¯¯¯¯¯¯¯PR=(λ+11)i+(λ2)j
=λi+(λ2)j
They don't form a triangle.
Hence,¯¯¯¯¯¯¯¯PQ+¯¯¯¯¯¯¯¯¯QR¯¯¯¯¯¯¯¯PR
i(λ+33)+j(λ3+1)i(λ)+j(λ2)
i(λ)+j(λ3+1)i(λ)+j(λ2)
i(λ)+j(λ2)i(λ)+j(λ2)
L.H.S and R.H.S can't be equal,
Both have to be zero,then¯¯¯¯¯¯¯¯PR=¯¯¯0,
hence no side and hence,no triangle will be possible.$
λ=0,2.
But sine λ can't be equal to 0 and 2 at the same time.
Hence,there is no value of λ for which the given points can't form a triangle.

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