The equations of two circles, C1 and C2 are x^2+y^2+7 x-10 y+31=0 and x^2+y^2-x-6 y+3=0 respecti

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The equations of two circles, C1 and C2 are x^2+y^2+7 x-10...

The equations of two circles, $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ are $x^{2}+y^{2}+7 x-10 y+31=0$ and $x^{2}+y^{2}-x-6 y+3=0$ respectively. Find the equation of the circle, $\mathrm{C}_{3}$, which passes through the intersection of the circles $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ and has its center on the line $\ell$ with equation $\mathrm{x}-\mathrm{y}-2=0$.

The equations of two circles, $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ are $x^{2}+y^{2}+7 x-10 y+31=0$ and $x^{2}+y^{2}-x-6 y+3=0$
respectively. Find the equation of the circle, $\mathrm{C}_{3}$, which passes through the intersection of the circles $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ and has its center on the line $\ell$ with equation $\mathrm{x}-\mathrm{y}-2=0$.

Key Concepts

Geometric Conditions from Center Placement

When additional conditions are imposed on a circle, such as its center lying on a specific line, these conditions provide equations that relate the center coordinates. Incorporating these constraints into the standard form of the circle's equation allows one to solve for unknown parameters. This approach is essential in problems where the circle must satisfy both intersection and positional requirements.

Family of Circles Through Intersection Points

A circle that passes through the intersection points of two given circles can be expressed as a linear combination of the equations of those circles. This concept is based on the fact that any circle through the common points of two circles lies in a family defined by the equations of the two circles. This method reduces the problem of finding such a circle to determining the appropriate combination constant that satisfies additional conditions.

Completing the Square

This technique is used to rewrite a general quadratic equation of a circle into its standard form. By grouping the x?terms and y?terms and adding appropriate constants, one can express the equation in the form (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This conversion is fundamental in coordinate geometry since it makes properties such as the center and radius immediately apparent.

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