Question

If the lines $2\cos \theta +3\sin \theta =\frac{5}{r}$, $3\cos \theta +4\sin \theta =\frac{7}{r}$ and $7\cos \theta +9\sin \theta =\frac{k}{r}$ are concurrent then $k=$

• A. $16$
• B. $15$
• C. $2$
• D. $-2$

Answer 1

The correct option is A $16$

Given lines are concurrent
$2\cos \theta +3\sin \theta =\frac{5}{r}$

$3\cos \theta +4\sin \theta =\frac{7}{r}$

$7\cos \theta +9\sin \theta =\frac{k}{r}$
$\Rightarrow \left|\begin{array}{ccc}2& 3& -5\\ 3& 4& -7\\ 7& 9& -k\end{array}\right|=0$

$\Rightarrow k=16$