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Q.
If a circle passes through the points where the lines $3 k x-2 y-1=0$ and $4 x-3 y+2=0$ meet the coordinate axes then $k=$
Conic Sections
Solution:
The line $3 k x-2 y-1=0$ meets $x$-axis and $y$-axis at $A\left(\frac{1}{3 k}, 0\right)$ and $B\left(0,-\frac{1}{2}\right)$, respectively.
Also, the line $4 x-3 y+2=0$ cuts $x$-axis and $y$-axis at $C\left(-\frac{1}{2}, 0\right)$ and $D\left(0, \frac{2}{3}\right)$, respectively.
Since the four points are concyclic, we have
$ O B \times O D=O A \times O C$
$\Rightarrow (1 / 2)(2 / 3)=(1 / 3 k)(1 / 2)$
$\Rightarrow k=1 / 2$
