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Subject: "An Interesting Property of Ellipse"     Previous Topic | Next Topic
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rama1977
Member since Sep-1-05
Sep-01-05, 08:58 AM (EST)
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"An Interesting Property of Ellipse"
 
   I found one onteresting property of Ellipse. Let P(x.y) be the any point on ellipse. Let C & D be the endpoints of Minor axis. Then the product of slopes of the lines joining PC and PD is always constant and is equal to (-a2/b2)where a is Semi major axis and b is Semi minor axis.

The theoretical background may please be elaborated.

V.Venkat Ramayya


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  Subject     Author     Message Date     ID  
An Interesting Property of Ellipse rama1977 Sep-01-05 TOP
  RE: An Interesting Property of Ellipse JJ Sep-05-05 1
     RE: An Interesting Property of Ellipse V Venkat Ramyya Sep-21-05 3
  RE: An Interesting Property of Ellipse JJ Sep-05-05 2
  RE: An Interesting Property of Ellipse Owen Oct-25-05 4
  RE: An Interesting Property of Ellipse mpdlc Oct-26-05 5
     RE: An Interesting Property of Ellipse Owen Oct-27-05 6

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JJ
guest
Sep-05-05, 06:56 AM (EST)
 
1. "RE: An Interesting Property of Ellipse"
In response to message #0
 
   It is very easy.
Slopes are :
(y-b)/x and (y+b)/x
product = (y² - b²)/x²
Equation of ellipse : (x/a)² +(y/b)² = 1
Hence b²(x/a)² + y² = b²
y² - b² = -x²b²/a²
product = (y²-b²)/x² = -b²/a²


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V Venkat Ramyya
guest
Sep-21-05, 07:13 AM (EST)
 
3. "RE: An Interesting Property of Ellipse"
In response to message #1
 
   Dear Sir!
I have already proved it. In fact it happens with end points of major axis also. Kindly explain the theoretical background.
Venkat Ramayya


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JJ
guest
Sep-05-05, 06:56 AM (EST)
 
2. "RE: An Interesting Property of Ellipse"
In response to message #0
 
   impossible to write properly the equations :
Some signs + and - always disappear.


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Owen
guest
Oct-25-05, 09:33 PM (EST)
 
4. "RE: An Interesting Property of Ellipse"
In response to message #0
 
   I think that if you use a hyperbola instead of an ellipse and let C and D be the two vertices, then the products of the slopes will be b^2/a^2. It makes sense that this value is positive whereas for the ellipse it was negative.


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mpdlc
guest
Oct-26-05, 02:01 PM (EST)
 
5. "RE: An Interesting Property of Ellipse"
In response to message #0
 
   The property you found for the ellipse in my opinion is an obvious consequence of the following properties.


1) Given a semicircunference of diameter 2a any ordinate Y will divide the diameter in two segment L and (2a-L).

Since Y is the heigh of the straight triangle with the diameter as base.
We can write equation Y^2 = L(2a- L)

or better this way (Y/L)(Y/(2a-L)) =1. ( i )

It is easy to recognize the two factor of the first member as the slope formed by the cords : legs of the straight triangle; with the diameter : the hypothenuse.

So we conclude that in a circuference the product of the two slopes equal to one

2) You probably remember an ellipse can be obtain by an affine contraction of the circunference so the parametric equation for a circunference of radius a we write :

X = a cos(phi) ; Y = a sin(phi)

meanwhile the parametric equation of an ellipse it will be written

x = a cos(phi) ; y = b sin(phi)

Indeed we have applied an scale reduction equal to (b/a) for the ordinate Y of the cicunference, so for ellipse the ordinate y become equal to y=(b/a)Y .

3) By multiplying the equation ( i ) by (b/a) twice, we shall get

(y/L)(y/(2a-L)) = (b/a)^2 in agreement with your result.

The sign different is due to the way the angle slope had being taken. Indeed the equation ( i ) should equal to -1 instead of 1 if both angles were mesured with the same convention (clockwise or couterclockwise).


This explanation does not diminish your finding, I never heard about it.

Now you can brag ....in case of noboby have done before.... to father another definition of the ellipse somewhat like:

Given a segment, the ellipse is the locus of points for which the product of the two slopes referred to the endpoints of the segment is a constant.

I apologize for my English, it is not my mother language, so you can improve the definition.



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Owen
guest
Oct-27-05, 06:34 AM (EST)
 
6. "RE: An Interesting Property of Ellipse"
In response to message #5
 
   Regarding your definition:
"Given a segment, the ellipse is the locus of points for which the product of the two slopes referred to the endpoints of the segment is a constant."

I think it would read better if the constant were given first, along with the segment. Also, I believe the constant must be positive; the locus will be a hyperbola if the constant is negative. How about this?

1) Given a segment AB and constant c > 0 (c < 0), the set of points X for which the product of the slopes of line AX and BX equals c is an ellipse (hyperbola).

I'm not that thrilled with this definition, either. I think a definition should start (in this case) with "An ellipse is ...", rather than end with "...is an ellipse" unless one introduces parameters in the beginning that become part of the term being defined.

For example, one could define a (specific) circle as follows.
"Given a point P in a plane and radius r, the locus of points at distance r from P is said to be the circle with center P and radius r."

My definition in (1) doesn't do this, as no attributes of the ellipse are referred to.

How about this?

2) An ellipse (hyperbola) is the set of points X for which the product of the slopes of the two lines through X and two fixed points is a positive (negative) constant.

I'm still not sure I have it right, as mathematical grammar is so subtle. Of course, here, one must insert mental parenthesis and commas appropriately to read it as

An ellipse is {X: (slope of XA)(slope of XB) = c} for some points A and B and constant c>0. Then it is clear that A, B, and c are fixed ahead of time and the ellipse will depend on them. Qualifiers are one of the most difficult aspects of definitions for many students, I believe.

When one encounters the first X in (2) it'sounds like X is a set, not a point in a set, and it isn't clear until one reaches the second X that X must be a point. In my view this makes (2) not a very good definition either. Perhaps you have similar issues in the Spanish language.

It would be nice not to mention X at all. You didn't refer to X in your definition, but I didn't think the phrase "the product of the two slopes referred to the endpoints of the segment is a constant" was clear.

Well, what I thought would be a 2-minute post giving a clean definition turned into something else entirely. Sorry about that. Someone else want to give the definition a try?


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