Back to Analytic Geometry repository
Analytic Geometry2D

The Parabola - Theory & Concepts - Analytic Geometry Parabola Focus Directrix

Equations of parabolas, vertex, focus, directrix, and applications.

Open the complete lesson

Parabola: Focus & Directrix Explorer

Parabola parameters

Vertex h (x-center)0
Vertex k (y-center)-2
Focal parameter p (focus distance)2

Test point on parabola

Point P x-coordinate (x_P)4.0

Geometric Locus Distance Verification

Equation & Coordinates
(x−h)2=4p(y−k)(x - h)^2 = 4p(y - k)
Focus: F(h,k+p)=F(0,0.0)F(h, k + p) = F(0, 0.0)
Directrix: y=k−p  ⟹  y=−4.0y = k - p \implies y = -4.0
Point: P(xP,yP)=P(4.0,0.00)P(x_P, y_P) = P(4.0, 0.00)
1. Distance to Focus (d₁)
d1=(xP−h)2+(yP−yF)2=(4.0−0)2+(0.00−0.0)2=4.000d_1 = \sqrt{(x_P - h)^2 + (y_P - y_F)^2} = \sqrt{(4.0 - 0)^2 + (0.00 - 0.0)^2} = 4.000
2. Distance to Directrix (d₂)
d2=∣yP−ydir∣=∣0.00−(−4.0)∣=4.000d_2 = |y_P - y_{\text{dir}}| = |0.00 - (-4.0)| = 4.000
d₁ = d₂ = 4.000 units (Verified!)
DIRECTRIX (y = -4.0)Focus F (0, 0)F(0, 0.0)Vertex V (0, -2)V(0, -2)Point P (4.0, 0.0)P(4.0, 0.0)

Move the "Point P x-coordinate" slider on the left to trace the parabola

Related simulations

Introduction to Analytic Geometry - Theory & Concepts - Analytic Geometry Distance SlopeIntroduction to Analytic Geometry - Theory & Concepts - Analytic Geometry Section FormulaIntroduction to Analytic Geometry - Theory & Concepts - Analytic Geometry Polygon AreaIntroduction to Analytic Geometry - Theory & Concepts - Analytic Geometry Conic Section Cutter