Question

The lines $a\cos \theta +2a\sin \theta +\frac{1}{r}=0,b\cos \theta +3b\sin \theta +\frac{1}{r}=0$ and $c\cos \theta +4c\sin \theta +\frac{1}{r}=0$ are concurrent then $a,b,c$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. A.G.P

Answer 1

Answer: (C)

H.P.

Given line are concurrent
$a\cos \theta +2a\sin \theta +\frac{1}{r}=0$
$b\cos \theta +3b\sin \theta +\frac{1}{r}=0$
$c\cos \theta +4c\sin \theta +\frac{1}{r}=0$
$\Rightarrow \left|\begin{array}{ccc}a& 2a& 1\\ b& 3b& 1\\ c& 4c& 1\end{array}\right|=0$
$\Rightarrow ab+bc=2ac$
$\Rightarrow \frac{2}{b}=\frac{1}{a}+\frac{1}{c}$
Hence, $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in $AP$.
$a,b,c$ are in H.P.