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Question

If p+q+r=0, prove that
∣ ∣paqbrcqcrapbrbpcqa∣ ∣=pqr∣ ∣abccabbca∣ ∣

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Solution

By expansion
L.H.S.=pa(qra2p2bc)qb(q2acprb2)+re(pqc2r2ab)
=pqr(a3+b3+c3)abc(p3+q3+r3) ..(1)
Now p+q=r=0 p3+b3+c33abc)
Putting in (1), we get
Δ=pqr(a3+b3+c33abc)
Again R.H.S.
=pqr[a(a2bc)b(cab2)+c(c2ab)]
=pqr[a3+b3+c33abc]=L.H.S.


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