Question

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

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

Answer 1

Answer: (A)

A.P.

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