Hyperbola Equations in Coordinate Geometry
⭕General Form of a Hyperbola
Ax² - Cy² + Dx + Ey + F = 0
where A ≠ 0, C ≠ 0, and A and C have opposite signs for a real hyperbola.
Center
(-D/(2A), -E/(2C))
h = -D/(2A), k = -E/(2C)
Semi-axes
a² = discriminant/A
b² = -discriminant/C
Asymptotes
y - k = ±(b/a)(x - h)
Two diagonal lines
🎯Standard Form and Orientation
Horizontal Transverse Axis
(x - h)²/a² - (y - k)²/b² = 1
Center: (h, k)
Vertices: (h±a, k)
Foci: (h±c, k)
where c² = a² + b²
Vertical Transverse Axis
(y - k)²/a² - (x - h)²/b² = 1
Center: (h, k)
Vertices: (h, k±a)
Foci: (h, k±c)
where c² = a² + b²
⚡Eccentricity and Foci
Eccentricity (e)
e = √(1 + b²/a²)
or e = c/a
Range: e > 1
Always greater than 1 for hyperbolas
Foci Distance
c = √(a² + b²)
Distance from center to focus
Note: c > a always
Unlike ellipse where c < a
Focal Property
|d₁ - d₂| = 2a
Property: Difference of distances
From any point to foci is constant
Absolute difference
📐Asymptotes
Horizontal Hyperbola
y - k = ±(b/a)(x - h)
Equations:
- • y = k + (b/a)(x - h)
- • y = k - (b/a)(x - h)
- • Pass through center (h, k)
Vertical Hyperbola
y - k = ±(a/b)(x - h)
Properties:
- • Hyperbola approaches but never touches
- • Slopes are ±a/b
- • Form a rectangle with vertices
🔄Special Cases
Rectangular Hyperbola
a = b
Equation: xy = c²/2
Asymptotes perpendicular
45° rotated form
Centered at Origin
x²/a² - y²/b² = 1
Center: (0, 0)
Simplest form
Most common in problems
Conjugate Hyperbola
-x²/a² + y²/b² = 1
Property: Swapped axes
Same asymptotes
Perpendicular transverse axes
📝Parametric Form
Horizontal Hyperbola
x = h + a sec(t)
y = k + b tan(t)
- • Uses secant and tangent
- • Parameter t is angle
- • Traces one branch at a time
Alternative Form (Hyperbolic)
x = h + a cosh(t)
y = k + b sinh(t)
- • Uses hyperbolic functions
- • Natural for hyperbolas
- • Traces entire branch smoothly
🔬 Real-World Applications
- • Navigation: LORAN and GPS positioning systems
- • Physics: Particle trajectories and orbits
- • Architecture: Cooling towers and structural design
- • Optics: Hyperbolic mirrors and lenses
- • Astronomy: Comet and spacecraft trajectories
🧮 Problem-Solving Tips
- • Check A and C have opposite signs
- • Identify transverse axis orientation
- • Remember: c² = a² + b² (not subtraction!)
- • Asymptotes pass through center
- • Eccentricity e > 1 always
- • Focal property uses difference, not sum