Parabola Equations in Coordinate Geometry
⭕General Quadratic Form
y = ax² + bx + c
where a ≠ 0 (if a = 0, it's a line, not a parabola)
Vertex
x = -b/(2a)
y = c - b²/(4a)
Direction
a > 0: opens up
a < 0: opens down
Axis of Symmetry
x = -b/(2a)
Vertical line through vertex
🎯Standard Form and Orientation
Vertical Parabola
(x - h)² = 4p(y - k)
Vertex: (h, k)
Focus: (h, k+p)
Directrix: y = k-p
Opens: Up if p > 0, Down if p < 0
Horizontal Parabola
(y - k)² = 4p(x - h)
Vertex: (h, k)
Focus: (h+p, k)
Directrix: x = h-p
Opens: Right if p > 0, Left if p < 0
⚡Focus and Directrix Properties
Definition
d₁ = d₂
Property: Equal distances
Any point on parabola is equidistant from focus and directrix
Defining property
Focal Parameter (p)
p = distance
From vertex to focus
Sign: Determines direction
|p| = focal length
Latus Rectum
Length = 4|p|
Definition: Chord through focus
Perpendicular to axis
Measures "width" at focus
📐Vertex Form
Vertical Parabola
y = a(x - h)² + k
Vertex: (h, k)
- • a > 0: opens upward
- • a < 0: opens downward
- • |a| affects "width"
- • Relationship: a = 1/(4p)
Converting Forms
Standard → Vertex:
Complete the square
Vertex → Standard:
p = 1/(4a)
Both forms useful for different problems
🔄Special Cases and Forms
Vertex at Origin
x² = 4py
Vertex: (0, 0)
Simplest form
Most common in problems
Simple Quadratic
y = x²
Properties: a = 1, p = 1/4
Focus: (0, 1/4)
Directrix: y = -1/4
General Conic
Ax² + Bxy + Cy²...
Parabola if: B² = 4AC
Discriminant test
Identifies conic type
📝Parametric Form
Vertical Parabola (x² = 4py)
x = 2pt
y = pt²
- • Parameter t is real number
- • Simple to compute
- • Traces entire parabola
Horizontal Parabola (y² = 4px)
x = pt²
y = 2pt
- • x and y roles swapped
- • Used for projectile motion
- • t often represents time
📊Tangent and Normal Lines
Tangent at Point (x₁, y₁)
For x² = 4py:
xx₁ = 2p(y + y₁)
• Point must be on parabola
• Unique tangent at each point
• Slope = x₁/(2p)
Reflection Property
Key Property:
Ray parallel to axis reflects through focus
• Used in satellite dishes
• Car headlights design
• Solar concentrators
🔬 Real-World Applications
- • Physics: Projectile motion trajectories
- • Engineering: Satellite dishes and antennas
- • Architecture: Arches and bridge designs
- • Optics: Parabolic mirrors and reflectors
- • Automotive: Headlight reflectors
- • Sports: Basketball shot trajectories
🧮 Problem-Solving Tips
- • Identify vertex position first
- • Determine axis orientation (vertical/horizontal)
- • Find p from coefficient or focus/directrix
- • Vertex form easier for graphing
- • Standard form shows focus clearly
- • Complete the square to convert forms
- • Check opening direction (sign of a or p)