Question

If the lines $a\cos \theta +b\sin \theta +\frac{c}{r}=0$, $b\cos \theta +c\sin \theta +\frac{a}{r}=0$ and $c\cos \theta +a\sin \theta +\frac{b}{r}=0$ are concurrent then

• A. $a^3+b^3+c^3=3abc$
• B. $a^3+b^3+c^3=0$
• C. $a^2+b^2+c^2=0$
• D. $a^3+b^3+c^3+3abc=0$

Answer 1

The correct option is A $a^3+b^3+c^3=3abc$

Given equation of lines
$a\cos \theta +b\sin \theta +\frac{c}{r}=0$
$b\cos \theta +c\sin \theta +\frac{a}{r}=0$
$c\cos \theta +a\sin \theta +\frac{b}{r}=0$
Since, the lines are concurrent
$\Rightarrow \left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=0$
$\Rightarrow a(bc-a^2)-b(b^2-ac)+c(ab-c^2)$
$\Rightarrow a^3+b^3+c^3=3abc$