Question

If $a,b,c$ are the sides of a triangle $ABC$ such that $x^{2}2(a+b+c)x+3λ(ab+bc+ca)=0$ has real roots, then

• A. $λ<\frac{4}{3}$
• B. $λ>\frac{5}{3}$
• C. $λ∈(\frac{4}{3},\frac{5}{3})$
• D. $λ∈(\frac{1}{3},\frac{5}{3})$

Answer 1

Answer: (A)

Since, roots are real, therefore $Dâ‰\yen 0$
$⇒4(a+b+c{)}^2−12λ(ab+bc+ca)â‰\yen 0$
$⇒(a+b+c{)}^2â‰\yen 3λ(ab+bc+ca)$
$⇒(a^2+b^2+c^2)â‰\yen (ab+bc+ca)(3λ−2)$
$⇒3λ−2≤\frac{a^2+b^2+c^2}{ab+bc+ca}...(i)$
Also, $\cos A=\frac{b^2+c^2−a^2}{2bc}<1⇒b^2+c^2−a^2<2bc$
Similarly, $c^2+a^2−b^2<2ca$
and $a^2+b^2−c^2<2ab$
$⇒a^2+b^2+c^2<2(ab+bc+ca)$
$⇒\frac{a^2+b^2+c^2}{ab+bc+ca}<\mathrm{2...}(ii)$
From Eqs. (i) and (ii), we get
$3λ−2<2⇒λ<\frac{4}{3}$