If the lines 2x+3y+1=0 and 3x-y-4=0 lie along diameters of a circle of circumference 10$\pi$, then the equation of the circle is 

Point of intersection of 2x+3y+1=0, 3x-y-4=0 is (1, -1)

$\Rightarrow$Centre of the circle=$\left(1, -1\right)$

Circumference =$10\pi \Rightarrow 2\pi r=10\pi \Rightarrow r=5.$

Equation of the circle is ${\left(x-1\right)}^2+{\left(y+1\right)}^2=5^2\Rightarrow x^2+y^2-2x+2y-23=0.$