Two circles x^2+y^2=6 and x^2+y^2-6 x+8=0 are given. Then...

Two circles $x^{2}+y^{2}=6$ and $x^{2}+y^{2}-6 x+8=0$ are given. Then the equation of the circle through their points of intersection and the point $(1,1)$ is
(a) $x^{2}+y^{2}-6 x+4=0$
(b) $x^{2}+y^{2}-3 x+1=0$
(c) $x^{2}+y^{2}-4 y+2=0$
(d) none of these

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Key Concepts

Family of Circles Passing Through Intersection Points

Any circle that passes through the points of intersection of two given circles can be represented as a linear combination of the equations of these circles. This method ensures that the resulting circle’s equation is satisfied at the intersection points since both original equations equal zero there.

Determining a Specific Circle Using an External Point

When a circle is required to pass through an additional specific point, this condition can be used to determine the free parameter in the general linear combination. Substituting the coordinates of the external point into the general equation allows for the calculation of the parameter, thus yielding the equation of the unique circle that satisfies all conditions.

Algebraic Manipulation in Coordinate Geometry

The process involves combining and rearranging quadratic equations representing circles, involving tasks such as completing the square, isolating variables, and simplifying expressions. Mastery of these algebraic techniques is essential for solving problems related to the intersections and common chords of circles.

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