A beautifully constructed inequality---IIT 2006

Since the given quadratic equation has real roots $\Rightarrow b^{2}-4ac\geq 0$ $\Rightarrow 4(a+b+c)^{2}-12\lambda (ab+bc+ca)\geq 0$ $\Rightarrow 4[(a+b+c)^{2}-3\lambda (ab+bc+ca)]\geq 0$ $\Rightarrow(a+b+c)^{2}-3\lambda (ab+bc+ca)\geq 0$ $\Rightarrow \lambda \leq \frac{(a+b+c)^{2}}{3(ab+bc+ca)}$ But we know that ,when a,b,c are sides of a triangle, $a^{2}+b^{2}+c^{2}\leq 2(ab+bc+ca)$ $\Rightarrow (a+b+c)^{2}\leq 4(ab+bc+ca)$ Therefore, $\lambda \leq \frac{(a+b+c)^{2}}{3(ab+bc+ca)}\leq \frac{4(ab+bc+ca)}{3(ab+bc+ca)}= \frac{4}{3}$ $\Rightarrow \lambda < \frac{4}{3}$ $\Rightarrow \sqrt{m^{2}+n^{2}}=\sqrt{4^{2}+3^{2}}=5$