If the equation of the incircle of an equilateral triangle is ${ x }^{ 2 }+{ y }^{ 2 }+4x-6y+4=0$, then the equation of the circumcircle of the triangle is

Given equation of the incirle is ${ x }^{ 2 }+{ y }^{ 2 }+4x-6y+4=0$

Its incenter is $(-2,3)$ and inradius $=\sqrt { 4+9-4 } =3$

Since in an equilateral triangle, the incenter and the circumcenter coincide,

$\therefore$ Circumcenter $=(-2,3)$

Also, in an equilateral triangle, circumradius $=2($ inradius$)$

$\therefore$ Circumradius $=2.3=6$

$\therefore$ The equation of the circumcircle is

${ \left( x+2 \right) }^{ 2 }+{ \left( y-3 \right) }^{ 2 }={ \left( 6 \right) }^{ 2 }\Rightarrow { x }^{ 2 }+{ y }^{ 2 }+4x-6y-23=0$