Circle Equations in Coordinate Geometry
⭕General Form of a Circle
ax² + ay² + 2gx + 2fy + c = 0
where a ≠ 0 and g² + f² - ac > 0 for a real circle.
Center
(-g/a, -f/a)
h = -g/a, k = -f/a
Radius
√(g² + f² - ac)/|a|
r = √(discriminant)/|a|
Condition
g² + f² - ac > 0
For real circle
🎯Standard Form and Transformations
Standard Form
(x - h)² + (y - k)² = r²
Center: (h, k)
Radius: r
Domain: [h - r, h + r]
Range: [k - r, k + r]
Conversion Formulas
General → Standard:
h = -g/a, k = -f/a
r² = (g² + f² - ac)/a²
Standard → General:
g = -ah, f = -ak
c = a(h² + k² - r²)
⚡Special Cases and Properties
Unit Circle
x² + y² = 1
Center: (0, 0)
Radius: 1
Used in: Trigonometry
Centered at Origin
x² + y² = r²
Center: (0, 0)
Radius: r
Simplest form
Point Circle
r = 0
When: g² + f² = ac
Result: Single point
Degenerate case
📐Distance and Tangent Properties
Distance from Point to Circle
d = |√((x₀-h)² + (y₀-k)²) - r|
Point (x₀, y₀):
- • Inside: distance < r
- • On circle: distance = r
- • Outside: distance > r
Tangent Line
At point (x₁, y₁) on circle:
(x₁-h)(x-h) + (y₁-k)(y-k) = r²
From external point:
Length = √((x₀-h)² + (y₀-k)² - r²)
🔄Circle Relationships
Two Circles
Distance between centers:
d = √((h₁-h₂)² + (k₁-k₂)²)
• Separate: d > r₁ + r₂
• External tangent: d = r₁ + r₂
• Intersecting: |r₁ - r₂| < d < r₁ + r₂
• Internal tangent: d = |r₁ - r₂|
• One inside other: d < |r₁ - r₂|
Circle and Line
Line: Ax + By + C = 0
Distance from center:
d = |Ah + Bk + C|/√(A² + B²)
• No intersection: d > r
• Tangent: d = r
• Two intersections: d < r
📝Alternative Forms
Parametric Form
x = h + r cos(t)
y = k + r sin(t)
where 0 ≤ t ≤ 2π
- • Useful for plotting points
- • Easy to animate
- • Parameter t represents angle
Polar Form (Centered at Origin)
r = constant
where r is the radius
- • Simplest in polar coordinates
- • All points equidistant from origin
- • θ can vary from 0 to 2π
🔬 Real-World Applications
- • Engineering: Wheel and gear design
- • Physics: Circular motion and orbits
- • Computer Graphics:Drawing curves
- • Navigation: Radar and GPS systems
- • Architecture: Arches and domes
🧮 Problem-Solving Tips
- • Check discriminant: g² + f² - ac > 0
- • Use standard form for geometric problems
- • Convert between forms as needed
- • Parametric form helpful for animations
- • Remember: center is (-g/a, -f/a)