Mathematics - Straight Lines Question with Solution | TestHub

The two lines will be identical if there exists some real number $k$, such that $b^3-c^3=k(b-c)$, $c^3-a^3=k(c-a)$ and $a^3-b^3=k(a-b)\Rightarrow b-c=0$ or $b^2+c^2+bc=k;c-a=0$ or $c^2+a^2+ca=k;a-b=0$ or $a^2+b^2+ab=k$; that is, $b=c$ or $c=a$ or $a=b$. Next $b^2+c^2+bc=c^2+a^2+ca\Rightarrow b^2-a^2=c(a-b)$. Hence $a=b$ or $a+b+c=0$.