x² + y² + 6x – 8y + 21 = 0 is the equation of the incircle of an equilateral triangle, then the equation of the circumcircle of the triangle is

Concept:

The general equation for a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

Distance between two points (x₁, y₁) and (x₂, y₂) is given by,

d = \(\sqrt {{{\left( {{{\rm{x}}_2}{\rm{}} - {\rm{}}{{\rm{x}}_1}} \right)}^2}{\rm{}} + {\rm{}}{{\left( {{{\rm{y}}_2} - {{\rm{y}}_1}} \right)}^2}}\)

Calculation:

Equation of incircle x² + y² + 6x – 8y + 21 = 0

⇒ x² + 6x + 9 – 9 + y²– 8y + 16 – 16 + 21 = 0

⇒ (x² + 6x + 9) + (y²– 8y + 16) – 9 -16 + 21 = 0

⇒ (x + 3)² + (y - 4)² = 4

∴ Centre (h, k) = (-3, 4) and radius of incircle = 2

From fig, sin 30° = OD/AO

⇒ AO = 2 × 2 = 4                               (∵ sin 30° = 1/2)

AO is the radius of circumcircle

∴ Equation of circumcircle

= (x + 3)² + (y - 4)² = (4)²

= x² + 6x + 9 + y²– 8y + 16 =16

⇒ x² + 6x + 9 + y²– 8y = 0

⇒ x² + y² + 6x – 8y + 9 = 0

Hence, option (1) is correct.