Question

If the lines $2x-3y+\lambda =0$, $3x-4y-13=0$ and $8x-11y-33=0$ are concurrent, then the value of $|\lambda |$ is

Answer 1

The given lines are
$2x-3y+\lambda =0,3x-4y-13=0$ and $8x-11y-33=0$.

Let $(h,k)$ be the point of concurrency,
Now, solving
$3h-4k-13=0$
$8h-11k-33=0$
We get
$h=11,k=5$
Putting this in third line equation, we get
$2\left(11\right)-3\left(5\right)+\lambda =0$
$\Rightarrow \lambda =-7\therefore |\lambda |=7$


Alternate method:
All three lines are concurrent if
$\left|\begin{array}{ccc}2& -3& \lambda \\ 3& -4& -13\\ 8& -11& -33\end{array}\right|=0\Rightarrow 2(132-143)+3(-99+104)+\lambda (-33+32)=0$
$\Rightarrow \lambda =-7\therefore |\lambda |=7$