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Let Delta = |{:(-bc,,b^(2)+bc,,c^(2)+bc),(a^(2)+ac,,-ac,,c^(2)+ac),(a^(2)+ab,,b^(2)+ab,,-ab):}| and the equation px^(3) +qx^(2) +rx+s=0 has roots a,b,c where a,b,c in R^(+) The value of Delta is |
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Answer» `le 9r^(2)//p^(2)` `Delta = |{:(-bc,,ab+AC,,ac+ab),(ab+bc,,-ac,,bc+ab),(ac+bc,,bc+ac,,-ab):}|` Nowusing `C_(2) to C_(2)-C_(1)" and " C_(3) to C_(3) -C_(1) ` and thentaking(ab+bc+ca) common from `C_(2) " and" C_(3)` we get `Delta =|{:(-bc,,1,,1),(ab+bc,,-1,,0),(ac+bc,,0,,-1):}|xx (ab +bc +ca)^(2)` Nowapplying `R_(2) to R_(2) +R_(1)` we get `Delta = |{:(-bc,,1,,1),(ab,,0,,1),(ac+bc,,0,,-1):}| (ab+bc+ca)^(2)` Expandingalong `c_(2)` we get `Delta =(ab+bc+ca)^(2)[ac+bc+ca)^(2)` `=(ab+bc+ca)^(2)` `=(r//p)^(3) =r^(3)//p^(3)` Now givena,b,care allpositivethen `A.M ge G.M.` `rArr (ab+bc+ac)/(3) ge (abxx bcxx ac)^(1//3)` `" or" (ab+bc+ac)^(3) ge 27A^(2)b^(2)c^(2)` `"or" (ab+bc+ca)^(3) ge 27(s^(2)//p^(2))` if `Delta =27` thenab+bc+ca =3 and GIVENTHAT `a^(2) +b^(2)+c^(2)=3` From `(a+b+c)^(2) =a^(2) +b^(2)+c^(2) +2 (ab+bc+ca)` we have `a+b+c = ne 3` `rArr a+b+c =3` `rArr 3p+q=0` |
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