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Question
If (a3a−1,a2−3a−1),(b3b−1,b2−3b−1) and(c3c−1,c2−3c−1) 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. ___
If (a3a−1,a2−3a−1),(b3b−1,b2−3b−1) and(c3c−1,c2−3c−1) 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.
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Solution
Let the equation of line on which these three points lie be
lx + my + n = 0
and the point (t3t−1,t2−3t−1) lie on the line where t = a, b, c
l(t3t−1)+m(t2−3t−1)+n=0
lt3+m(t2−3)+n(t−1)=0 t3l+t3m+tn−3m−n=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
Let the equation of line on which these three points lie be
lx + my + n = 0
and the point (t3t−1,t2−3t−1) lie on the line where t = a, b, c
l(t3t−1)+m(t2−3t−1)+n=0
lt3+m(t2−3)+n(t−1)=0 t3l+t3m+tn−3m−n=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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