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If (a3a1,a23a1),(b3b1,b23b1) and(c3c1,c23c1) are collinear and α(abc)+β(a+b+c)=γ(ab+bc+ca), where α, β, γ ϵ N, then find the least value of α+β+γ. ___


Solution

Let the equation of line on which these three points lie be

lx + my + n = 0

and the point (t3t1,t23t1) lie on the line where t = a, b, c

l(t3t1)+m(t23t1)+n=0

lt3+m(t23)+n(t1)=0 t3l+t3m+tn3mn=0

If a, b, c are the roots of given equation then

a+b+c=ml         . . . (i)

ab+bc+ca=nl      . . .. (ii)

abc=3ml+nl    .  . . (iii)

using (i), (ii) and (iii) we get

abc=3(a+b+c)+ab+bc+ca

abc+3(a+b+c)=ab+bc+ca

   α+β+γ=1+3+1=5

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