Circum circle of an equilateral triangle ABC is x^2+y^2+2 gx+2 fy=0. Then its incircle is (a) x^

step 1 In this problem of conic section we have given that circumcircicle of an equilateral triangle A

Circum circle of an equilateral triangle ABC is x^2+y^2+2...

Circum circle of an equilateral triangle $\mathrm{ABC}$ is $\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{gx}+2 \mathrm{fy}=0$. Then its incircle is (a) $x^{2}+y^{2}+2 g x+2 f y=0$ (b) $4\left(x^{2}+y^{2}\right)+8 g x+8 f y+3 g^{2}+3 f^{2}=0$ (c) $x^{2}+y^{2}+8 g x+8 f y+3 g^{2}+3 f^{2}=0$ (d) $x^{2}+y^{2}+2 g x+2 f y+3 g^{2}+3 f^{2}=0$

Circum circle of an equilateral triangle $\mathrm{ABC}$ is $\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{gx}+2 \mathrm{fy}=0$. Then its incircle is
(a) $x^{2}+y^{2}+2 g x+2 f y=0$
(b) $4\left(x^{2}+y^{2}\right)+8 g x+8 f y+3 g^{2}+3 f^{2}=0$
(c) $x^{2}+y^{2}+8 g x+8 f y+3 g^{2}+3 f^{2}=0$
(d) $x^{2}+y^{2}+2 g x+2 f y+3 g^{2}+3 f^{2}=0$

Key Concepts

Relationship between Circumradius and Inradius in an Equilateral Triangle

In an equilateral triangle, the circumradius R and the inradius r are related by the formula r = R/2. This relationship is a consequence of the symmetry of the triangle and links the circle passing through the vertices with the circle tangent to the sides.

Incircle

The incircle of a triangle is the circle that is tangent to all three sides of the triangle. In an equilateral triangle, due to the symmetry, the incenter (the center of the incircle) coincides with the circumcenter, meaning the incircle is centered at (-g, -f) as well.

Standard Equation of a Circle

A circle can be defined by the equation (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Expanding this equation leads to an expression that can be compared with given forms to determine the center and the radius, which is essential for understanding and deriving circle equations in coordinate geometry.

Equilateral Triangle

An equilateral triangle has all sides of equal length, which implies that its angles are all 60° and its various centers (circumcenter, incenter, centroid, and orthocenter) coincide. This property simplifies many geometric calculations, including the determination of the incircle and circumcircle equations.

Circumcircle

The circumcircle of a triangle is the unique circle that passes through all three vertices of the triangle. Its center, called the circumcenter, is equidistant from the vertices. For a given circle equation in the form x² + y² + 2gx + 2fy = 0, the center is at (-g, -f) and the radius is ?(g² + f²).

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