Question

If $a,b,c$ are in A.P. then the straight lines $ax+2y+1=0$, $bx+3y+1=0$ and $cx+4y+1=0$

• A. are concurrent
• B. from a triangle
• C. are parallel
• D. Can't say

Answer 1

The correct option is A are concurrent

$a,b,c$ are in $A.P$ and line are

$ax+2y+1=0$

$bx+3y+1=0$

$cx+4y+1=0$

For lines to be concurrent; $\Delta =0$

i.e., $\left|\begin{array}{ccc}a& 2& 1\\ b& 3& 1\\ c& 4& 1\end{array}\right|$

$L.H.S=\left|\begin{array}{ccc}a& 2& 1\\ b& 3& 1\\ c& 4& 1\end{array}\right|$

$R_2\to R_2-R_1$ & $R_3\to R_2-R_1$

$\Rightarrow \Delta =\left|\begin{array}{ccc}a& 2& 1\\ d& 1& 0\\ 2d& 2& 0\end{array}\right|$ where $d=$ common difference of $A.P$

$\Rightarrow \Delta =1\left\{2\right(d)-1(2d\left)\right\}-0+0$ [Expansion along third column]

$\Rightarrow \Delta =0$

$\Rightarrow$ Lines are concurrent