History

Coordinate methods made circle geometry algebraic: geometric questions about tangency and intersection can be solved by equations. This approach grew from analytic geometry.

Checkpoints

Find center and radius of \((x+4)^2+(y-1)^2=36\).

Answer: Center \((-4,1)\), radius \(6\).

Write equation of circle with center \((2,-3)\), radius \(7\).

Answer: \((x-2)^2+(y+3)^2=49\).

Is point \((5,1)\) on circle \((x-1)^2+(y-1)^2=16\)?

Answer: Yes, \((5-1)^2+(1-1)^2=16\).

Convert \(x^2+y^2-6x+4y-12=0\) to standard form.

Answer: \((x-3)^2+(y+2)^2=25\).

A line through tangent point has slope 2. What is slope of radius to that point?

Answer: \(-\frac12\).

Applications

• Engineering: circular component alignment and tolerance checks.
• Computer graphics: collision and boundary detection.
• Robotics: turning path and sensor range modeling.

References

• Standard high-school coordinate geometry materials on circles and tangents.
• Analytic geometry references on conic sections (intro level).