The number of possible straight lines passing through $\left(2,3\right)$ and forming a triangle with the coordinate axes , whose area is 12 sq. units, is 

The equation of any line through $\left(2,3\right)$ is 

$\mathrm{y}-3=\mathrm{m}\left(\mathrm{x}-2\right)$

or $\mathrm{y}=\mathrm{mx}-2\mathrm{m}+3$

From the figure, the area of $\mathrm{\Delta OAB}$ is $\pm 12,$ that is, 

$\frac{1}{2}\left(\frac{2\mathrm{m}-3}{\mathrm{m}}\right)\left(3-2\mathrm{m}\right)=\pm 12$

Taking positive sign, we get ${\left(2\mathrm{m}+3\right)}^2=0$. This gives one value of m as -3/2. Taking negative sign, we get

$4{\mathrm{m}}^2-36\mathrm{m}+9=0\left(\mathrm{D}>0\right)$

This is a quadratic in m which gives two values of m. Hence, three straight lines are possible.