Mathematics 5

Conics

Applications

# Mathematics Conics Ellipse

### 1. Ellipse: Construction:

An ellipse is defined as:

dist (P,F) + dist(P,F') = 2a

P is any point on the curve. F and F' are the two focii of the ellipse; and "dist" is the distance between the two related points.

Let's express the above equation:
sqrt [(c - x)2 + (0 - y)2 ] + [(- c - x)2 + (0 - y)2]1/2 = 2a

[(c - x)2 + y2 ]1/2 + [(- c - x)2 + y2 ]1/2 = 2a

[c2 - 2cx + x2 + y2]1/2 = 2a - [c2 + 2cx + x2 + y2]1/2

c2 - 2cx + x2 + y2 = 4a2 - 4a [c2 + 2cx + x2 + y2]1/2 + c2 + 2cx + x2 + y2

a [c2 + 2cx + x2 + y2]1/2 = a2 + cx

a2 [c2 + 2cx + x2 + y2] = a4 + 2cx a2 + c2x2

a2 [c2 + x2 + y2] = a4 + c2x2

a2 [ x2 + y2] - c2x2 = a2[a2 - c2]

x2[a2 - c2] + a2 y2] = a2[a2 - c2]

x2 b2 + a2 y2 = a2b2
Then:
x2/ a2 + y2/ b2 = 1

x2/ a2 + y2/ b2 = 1

That is the equation of an ellipse.

## 2.1. Eccentricity of an ellipse

The eccentricity "e" of a conic is defined as the ratio :

e = dist(P,F)/dist(P,L)

Applied to the point P = A, and then to the point P = A', yields:

e = dist(A,F)/dist(A,L)
e = (a - c)/(d - a)   (1)

e = dist(A',F)/dist(A',L)
e = (a + c)/(d + a)   (2)

Solving for "e":
ed - ea = a - c
ed + ea = a + c

2ed = 2a, then: e = a/d, or
Subtracting (2) from (1) gives: -2 ea = - 2c, then: e = c/a

e = c/a , with: 0 < e < 1

## 2.2. Polar equation of en ellipse

Cosine law gives:
PF2 = r2 + c2 - 2rc cosθ
We have :
PL = d - rcosθ
dist(P,F) = e dist(P,L), yields:
sqrt[r2 + c2 - 2r c cosθ] = e(d - r cosθ)

[r2 + c2 - 2r c cosθ]1/2 = e(d - r cosθ)
r2 + c2 - 2r c cosθ = e2d2 + e2 r2 cos2 θ - 2rde2 cosθ
r2[1 - e2 cos2 θ] + 2r cosθ [ de2 - c] + c2 - e2d2 = 0
Since ed = a, ae = c, and b2 = a2 - c2 then:
r2[1 - e2 cos2 θ] = b2
r = b/[1 - e2 cos2 θ]1/2

r = b/[1 - e2 cos2 θ]1/2 =
[(a2 - c2)/(1 - e2 cos2 θ)]1/2

To simplify, lest' consider F = O, that is the focus is at the origin c = 0, therefore:

ro = dist(F,P) = e dist(P,L)
d - c = do

dist(P,L) = do - rocos θo; then:
ro = e[do - rocos θo]
ro[1 + e cos θo] = edo
ro = edo/[1 + e cos θo]

ro = edo/(1 + e cos θo)

 Web ScientificSentence

 chimie labs | scientific sentence | java | Perl | php | green cat | contact |