Coordinate geometry of circles

Coordinate geometry of circles combines algebra with geometric properties such as radius, center, tangents, and intersections.

Standard equation of a circle

A circle with center \((h,k)\) and radius \(r\) has equation:

\[ (x-h)^2+(y-k)^2=r^2. \]

If center is origin, this simplifies to \(x^2+y^2=r^2\).

Reading center and radius

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

• Center: \((3,-2)\)
• Radius: \(5\)

General form

A circle can also appear as:

\[ x^2+y^2+Dx+Ey+F=0. \]

Complete the square in \(x\) and \(y\) to convert to standard form.

Tangents and radii

• A tangent touches a circle at exactly one point.
• Radius to tangent point is perpendicular to the tangent.

If radius slope is \(m\), tangent slope is \(-\frac1m\) (when defined).

Point position relative to circle

For circle center \(C\), radius \(r\), and point \(P\):

• \(CP<r\): inside
• \(CP=r\): on circle
• \(CP>r\): outside

Line-circle intersections

Substitute line equation into circle equation.

• Two solutions: secant intersects twice.
• One solution: tangent.
• No real solution: no intersection.

Exam strategy

• Write center and radius before computation.
• Use exact values first; round at final step.
• For tangents, use radius-perpendicular fact explicitly.
• Check whether solutions satisfy geometric context.

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

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).