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March 29, 2009

Elliptical carrom board

devadutta @ 11:13 pm, GMT +0000 ( 1238368418 ) Play

Slightly different kind of a question.

Say you construct an elliptical carrom board and keep the striker on a focus point (see the diagram below) and strike it. Which point on the board is the striker surely going to pass through after rebound irrespective of the angle at which it was struck?

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Also, tell us which property of the ellipse lets you arrive at this.

p.s: Although it looks like a math question, it is very much quiz style.

Cracked by: Shrey Goyal , udupendra , Dibyo , Raghuvansh , u2001137 , Jean Valjean , Anand Shankar , Ps , sidsen , Rohan , madhur , Nakul , Krishna , yaksha , duriel , Neeraj , Poornima , Ananth , shrik , sand , p vs np , MGS and Ajay Parasuram.

Answer:

The striker will pass through the other focus.

p vs np has a crisp explaination of the reason

“This is so, because the normal at the point of intersection of the lines passing through the two focii on the perimeter of the curve, is the angular bisector of the angle formed. Hence, by rules of ideal reflectivity (incidence angle= reflected angle), it needs to pass through the other focus.”


26 Responses to “Elliptical carrom board”

  1. The other focus point.
    If a light source is placed at one focus of an elliptic mirror, all light rays on the plane of the ellipse are reflected to the second focus.This is a consequence of the total travel length being the same along any wall-bouncing path between the two foci.


  2. udupendra
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    Through the other focus, of course. I don’t know what ‘property’ you are referring to, but proof is simple –

    Striking a carrom is the same as reflection – so, if you consider the hypothetical flat mirror that is tangential to the ellipse at the point of impact, the incident ray and the reflected ray will have to make the same angle, which means that for a given horizontal line, by law of corresponding triangles, the sum of the lengths of the incident ray and reflected ray should be constant at any point on the ellipse. Which is nothing but the definition for the two foci for the ellipse.


  3. Dibyo
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    The other focus, because of the The Reflective Property of an Ellipse


  4. Raghuvansh
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    Considering this as a problem of reflection, and knowing that in an elliptical mirror, when light is cast on the mirror from one focus it converges after reflection to the other focus, I would say the striker will also take the other focus on its path.


  5. u2001137

    The other focus.

    – Tangent at any point P to an ellipse bisects the angle between the two generator lines (i.e. the lines joining P to the two focii)
    – During the process of reflection, angle of incidence = angle of reflection
    Hence the striker, when reflected from any point on the edge travels along the other generator line, i.e. it will definitely pass through the other focus


  6. The other focus.


  7. u2001137

    typo in previous comment….meant normal to the tagent bisects teh angle between two generators


  8. Anand Shankar

    eccentricity will make it pass through the other focus?


  9. Ps

    Second focus

    principle behind elliptical reflectors and the so called whisper chambers : “the total travel length being the same along any wall-bouncing path between the two foci.”

    Given that an ellipse is the locus of points the sum of whose distances from two given points (the foci) is constant, because of which the reflected path and the incidnet path make equal angles to the tangent (at any given point).

    Therefore, the striker will always bounce off the wall of the elliptical carrom board and pass the other focus of the ellipse.


  10. sidsen
    2

    the second foci


  11. Rohan
    1

    Based purey on intuition, I would say the other focus of the ellipse.. Is it called the reflective property?


  12. madhur
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    The other focus

    Reflection Property of Ellipse.


  13. Nakul

    it goes thru the other Focus. The property being the sum of the focal distances being equal to the major axis 2a.


  14. Krishna

    The other focus!

    An ellipse is the locus of all points for which the sum of distances from two(given) focii are the same.


  15. yaksha
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    the striker passes through the other foci.
    however i dont know the property as i usually used to sleep in AVS’s class ;)


  16. duriel
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    The other focus.

    Property: sum of distances from both foci to any point on perimeter is constant.


  17. Neeraj

    The striker will surely pass through the other focus of the ellipse and the property is the reflection property of the ellipse.


  18. Poornima

    The other focus?
    Is it because, the sum of distances from any point on the ellipse to the foci is equal?


  19. Ananth
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    The reflective property of an ellipse. An ellipse has 2 focal points. The striker will definitely go through the other focal point in the ellipse.


  20. shrik
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    Through the other focus! This is the principle of whisper chambers and elliptical mirrors. Not sure what the property itself would be called though…

    *google* *google*

    The tangent line always makes equal angles with the generator lines? If the inside of an ellipse is a mirror, a light ray leaving one focus will always pass through the other? Actually by definition, since the distance between the rebound point and the foci always remains constant, one could say that it’s not any property, but the very definition of an ellipse that explains this.


  21. sand

    Through the other focus…. its like reflection, optical elliptical mirrors (convex and concave) are constructed by this principle.


  22. p vs np
    1

    Aaw.. The striker shall pass through the other focus of the carrom board/ellipse.

    This is so, because the normal at the point of intersection of the lines passing through the two focii on the perimeter of the curve, is the angular bisector of the angle formed. Hence, by rules of ideal reflectivity (incidence angle= reflected angle), it needs to pass through the other focus.


  23. MGS

    it passes thru the other focal point.

    Because: Reflection Theory. Angle of Incidence = Angle of Reflection.

    http://cage.rug.ac.be/~hs/billiards/billiards.html


  24. Ajay Parasuram
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    The striker will pass through either one of the 2 foci (+/-ae,0)


  25. p vs np
    1

    @Yaksha
    Lol! :D


  26. Rithwik K

    The other focus

    An ellipse is the locus of all points for which the sum of distances from two(given) focii are the same


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