Question

Let $a,b,c$ be the lengths of sides of a scalene triangle. If the roots of the equation $x^2+2(a+b+c)x+3\lambda (ab+bc+ca)=0$ $(\lambda \in R)$ are real then

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

Answer 1

Answer: (A)

$\lambda <\frac{4}{3}$

Since roots of given quadratic are real, $D\ge 0$
$\Rightarrow (a+b+c{)}^2-3\lambda (ab+bc+ca)\ge 0$
$\Rightarrow \lambda \le \frac{1}{3}\frac{(a+b+c{)}^2}{ab+bc+ca}=\frac{2}{3}+\frac{1}{3}\frac{a^2+b^2+c^2}{ab+bc+ca}$
Given $a,b,c$ are sides of triangle $\Rightarrow a+b>c\Rightarrow ac+bc>c^2$
Similarly, $ab+bc>b^2$ and $ca+ab>a^2$
Using these $a^2+b^2+c^2<2(ab+bc+ca)$
Hence, $\lambda <\frac{2}{3}+\frac{2}{3}=\frac{4}{3}$.