Key Idea The intersection of two diameters lines are the centre of circle.
The intersection of the lines $2 x-3 y=5$ and $3 x-4 y=7$ is $(1,-1)$
which is the coordinates of the centre.
Also, area of circle $=154$
$= \pi r^{2}=154$
$\Rightarrow r^{2}=\frac{154}{22 / 7}=7^{2}$
$\Rightarrow r=7$
$\therefore $ Required equation of circle is
$(x-1)^{2}+(y+1)^{2}=7^{2}$
$\Rightarrow x^{2}+y^{2}-2 x+2 y=47$