The equation (x+y-6)(x y-3 x-y+3)=0 represents the sides of...

The equation $(x+y-6)(x y-3 x-y+3)=0$ represents the sides of a triangle then the equation of the circumcircle of the triangle is
(A) $x^{2}+y^{2}-5 x-9 y+20=0$
(B) $x^{2}+y^{2}-4 x-8 y+18=0$
(C) $x^{2}+y^{2}-3 x-5 y+8=0$
(D) $x^{2}+y^{2}+2 x-3 y-1=0$

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Key Concepts

Intersection of Lines

Finding the points of intersection of lines is a key technique in determining the vertices of a triangle formed by intersecting lines. These intersection points are crucial for further geometric constructions, such as identifying the circumcenter or establishing the circumcircle.

Equation of a Circle

The equation of a circle in the Cartesian plane is typically expressed in the form (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r its radius. This equation can be expanded and rearranged into the general quadratic form, which is useful for analyzing or comparing different circles.

Circumcircle

The circumcircle of a triangle is the unique circle that passes through all three vertices of the triangle. This concept is fundamental in geometry and is used to discuss properties such as angles and distances within the triangle, as well as relationships with other geometric figures.

Circumcenter

The circumcenter is the point at which the perpendicular bisectors of the sides of a triangle intersect. It serves as the center of the circumcircle and is equidistant from all three vertices. This property is crucial when determining the equation of the circumcircle.

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