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

x^{2} + y^{2} + 4 x + 6 y - 23 = 0

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x^{2} + y^{2} + 4 x - 6 y - 23 = 0

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x^{2} + y^{2} - 4 x - 6 y - 23 = 0

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Solution

The correct option is A $x^{2} + y^{2} + 4 x - 6 y - 23 = 0$
Given equation of the incirle is $x^{2} + y^{2} + 4 x - 6 y + 4 = 0$
Its incenter is $(- 2, 3)$ and inradius $= \sqrt{4 + 9 - 4} = 3$
Since in an equilateral triangle, the incenter and the circumcenter coincide,
$∴$ Circumcenter $= (- 2, 3)$
Also, in an equilateral triangle, circumradius $= 2 ($ inradius $)$
$∴$ Circumradius $= 2.3 = 6$
$∴$ The equation of the circumcircle is
${(x + 2)}^{2} + {(y - 3)}^{2} = {(6)}^{2} \Rightarrow x^{2} + y^{2} + 4 x - 6 y - 23 = 0$

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If the equation of the incircle of an equilateral triangle is x2+y2+4x−6y+4=0, then the equation of the circumcircle of the triangle is

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