How to get formula for focus and directrix

  • Context: Undergrad 
  • Thread starter Thread starter ilya.s
  • Start date Start date
  • Tags Tags
    Focus Formula
Click For Summary
SUMMARY

The discussion centers on deriving the formulas for the focus and directrix of a parabola. The key equation presented is (X - Xf)² + (Y - Yf)² = (Y - Yd)², which represents the principle that the distance from any point on the parabola to the focus equals the distance to the directrix. A suggested approach is to start with a simple case where the focus is at (0, p) and the directrix is y = -p, or alternatively, focus at (p, 0) with directrix x = -p. This foundational understanding allows for the derivation of the parabola's properties using the distance formula.

PREREQUISITES
  • Understanding of parabolic geometry
  • Familiarity with the distance formula in coordinate geometry
  • Basic knowledge of Cartesian coordinates
  • Ability to manipulate algebraic equations
NEXT STEPS
  • Study the derivation of the parabola's standard equation from the focus and directrix
  • Explore the properties of parabolas in conic sections
  • Learn about the applications of parabolas in physics and engineering
  • Investigate the relationship between parabolas and quadratic functions
USEFUL FOR

Students of mathematics, educators teaching conic sections, and anyone interested in the geometric properties of parabolas.

ilya.s
Messages
1
Reaction score
0
Hello everyone, this is an amazing forum! :)

I have a question: I have seen the formulas for focus/directrix of a parabola, but I cannot trust something unless I know how to get there.

How do I get the formulas of focus and directrix?

I got to an equation (X - Xf)^2 + (Y - Yf)^2 = (Y - Yd)^2 (distance between any point and focus equals to the distance between the same point and directrix). Where do I go from here?

Thank you!
 
Physics news on Phys.org
You might start with a simple case where the focus and directrix are equidistant from the origin and dirextrix parallel to an axis: Focus (0,p) and directrix y = -p or symmetrically, Focus (p,0), directrix x = -p.

See http://en.wikipedia.org/wiki/Parabola.
 
You can derive the formula using the distance formula.
 

Similar threads

Undergrad Converse of focus-directrix property of conic sections
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad How directrix for other than parabola?
  • · Replies 1 ·
Replies
1
Views
1K
MHB Find the equation of parabola with vertex (2, 1) and focus (1, -1)
  • · Replies 3 ·
Replies
3
Views
30K
MHB Find Parabola Given Focus & Directrix - Help Needed
  • · Replies 2 ·
Replies
2
Views
6K
Undergrad Determining distance from Focus to Point on Parabola
  • · Replies 1 ·
Replies
1
Views
3K
High School Parabola vs Hyperbola, why does a Hyperbola have two foci/curves?
  • · Replies 3 ·
Replies
3
Views
2K
MHB Find Equation of Parabola Given Focus & Directrix - Jose's Q&A
  • · Replies 1 ·
Replies
1
Views
2K
Arriving at parabola formula via distance formula
  • · Replies 4 ·
Replies
4
Views
2K
MHB Finding equation of parabola with focus and directrix
  • · Replies 5 ·
Replies
5
Views
14K
Find Equation of Ellipse | Eccentricity 2/3 | (2,0) Focus & x+y=0 Directrix
  • · Replies 4 ·
Replies
4
Views
3K