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+++ title = "Coordinate geometry of circles" description = "Find equations of circles, tangents, and intersections using coordinate methods and distance." template = "chapter.html" weight = 26 [extra] subject = "math" chapter_id = "coordinate-circles" strand = "Measures, Shape & Space" +++
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
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).