Ellipse Equations in Coordinate Geometry
⭕General Form of an Ellipse
Ax² + Cy² + Dx + Ey + F = 0
where A ≠ 0, C ≠ 0, A ≠ C, and both have the same sign for a real ellipse.
Center
(-D/(2A), -E/(2C))
h = -D/(2A), k = -E/(2C)
Semi-axes
a = √(discriminant/A)
b = √(discriminant/C)
Condition
D²/(4A²) + E²/(4C²) - F > 0
For real ellipse
🎯Standard Form and Properties
Standard Form
(x - h)²/a² + (y - k)²/b² = 1
Center: (h, k)
Semi-major axis: max(a, b)
Semi-minor axis: min(a, b)
Domain: [h - a, h + a]
Range: [k - b, k + b]
Conversion Formulas
General → Standard:
h = -D/(2A), k = -E/(2C)
a² = discriminant/A
b² = discriminant/C
discriminant = D²/(4A²) + E²/(4C²) - F
Standard → General:
A = 1/a², C = 1/b²
D = -2h/a², E = -2k/b²
F = (h²/a² + k²/b² - 1)
⚡Eccentricity and Foci
Eccentricity (e)
e = √(1 - b²/a²)
where a > b
Range: 0 < e < 1
e = 0 is a circle, e → 1 is very elongated
Foci Location
c = ae
or c = √(a² - b²)
If a > b: (h±c, k)
If b > a: (h, k±c)
Horizontal or vertical
Focal Property
d₁ + d₂ = 2a
Property: Sum of distances
From any point to both foci is constant
Defines the ellipse shape
📐Special Cases and Axes
Centered at Origin
x²/a² + y²/b² = 1
Center: (0, 0)
- • Simplest form
- • Aligned with axes
- • Most common in problems
Axis Orientation
If a > b:
Major axis is horizontal
If b > a:
Major axis is vertical
Larger denominator determines major axis direction
📝Alternative Forms
Parametric Form
x = h + a cos(t)
y = k + b sin(t)
where 0 ≤ t ≤ 2π
- • Useful for plotting points
- • Easy to animate
- • Parameter t represents angle
- • Traces entire ellipse once
Polar Form (Centered at Origin)
r = ab/√((b cos θ)² + (a sin θ)²)
Alternative form with focus at origin:
r = a(1-e²)/(1-e cos θ)
- • More complex than circle
- • Used in orbital mechanics
- • θ is the polar angle
🔄Tangent and Normal Properties
Tangent at Point (x₁, y₁)
Equation:
(x₁-h)(x-h)/a² + (y₁-k)(y-k)/b² = 1
• Point must be on the ellipse
• Perpendicular to radius at that point
• Unique tangent at each point
Reflection Property
Focal Property:
A line from one focus reflects off the ellipse and passes through the other focus
• Used in whispering galleries
• Basis for elliptical mirrors
• Important in acoustics and optics
🔬 Real-World Applications
- • Astronomy: Planetary orbits around the sun
- • Architecture: Whispering galleries and acoustic design
- • Engineering: Elliptical gears and machine parts
- • Medicine: Lithotripsy (kidney stone treatment)
- • Optics: Reflectors and lens design
🧮 Problem-Solving Tips
- • Check A ≠ C for it to be an ellipse
- • Use standard form for geometric problems
- • Identify major axis (larger denominator)
- • Calculate eccentricity to find shape
- • Remember: center is (-D/(2A), -E/(2C))
- • Verify discriminant > 0 for real ellipse