Expert: Paul Klarreich - 7/18/2007

Question
I heard two interesting conics question the other day and was curious if you could explain it to me:

An elliptipool is an elliptical pool table with only one pocket.  A pool shark places a ball on the table, hits it in what appears to be a random direction, and yet it bounces off the edge, falling directly into the pocket.  Explain why this happens.

Radio tower M2 is located 200 miles due west of radio tower M1.  The situation is illustrated in the figure shown, where a coordinate system has been superimposed.  Simultaneous radio signals are sent from each tower to a ship, with the signal from M2 received 500 microseconds before the signal from M1.  Assuming that radio signals travel at 0.186 mile per microsecond, determine the equation of the hyperbola on which the ship is located.

Thanks in advance!  

Answer
Questioner:   Whitney
Category:  Advanced Math
Private:  No
 
Subject:  Conics
Question:  I heard two interesting conics question the other day and was curious if you could explain it to me:

An elliptipool is an elliptical pool table with only one pocket.  A pool shark places a ball on the table,

>> Please -- we don't have those any more.  They are simply expert players.

hits it in what appears to be a random direction, and yet it bounces off the edge, falling directly into the pocket.  Explain why this happens.

Radio tower M2 is located 200 miles due west of radio tower M1.  The situation is illustrated in the figure shown, where a coordinate system has been superimposed.  Simultaneous radio signals are sent from each tower to a ship, with the signal from M2 received 500 microseconds before the signal from M1.  Assuming that radio signals travel at 0.186 mile per microsecond, determine the equation of the hyperbola on which the ship is located.

Thanks in advance
.............................................
Hi, Whitney,

I am not sure about that first question -- perhaps there is some picture that you couldn't send.  If:

-- the table is an ellipse, and
-- the ball is struck at the center, and
-- it is struck in the direction of either axis, major or minor, and
-- the pocket is NOT along that axis, then

the ball can only ricochet along that axis and never fall into the hole.

HOWEVER, there is an interesting property of an ellipse which this question may be attempting to illustrate and if:

-- the table is an ellipse, and
-- the ball is struck at one focus, and
-- the pocket is at the other focus, (see note) then

the ball will inevitably reflect off the 'wall' and enter the pocket at the other focus.  Furthermore, the length of time it takes for the ball to drop will always be the same, assuming the expert player always strikes the ball with the same force.

This is the 'listening gallery' effect.  You can find the derivation in any calculus book.

Note: I know that the pockets are usually on the wall, but I can't think of any other ellipse property that might be relevant.

...................................

An hyperbola (see note) is the locus of points P such that the DIFFERENCE between the distances PF1 and PF2, where F1 and F2 are fixed points, called the foci, is a constant 2a

Note: Yes, you say 'An hyperbola', not 'A hyperbola'.

In your coordinate system (you didn't send the diagram, but it isn't hard to reproduce) you have the center at the origin and your foci are at the points:

M1(100,0),  M2(-100,0).

The equation of your horizontal hyperbola (the foci are on a horizontal line) with its center at (0,0) is:
x^2     y^2
----- - ----- = 1
a^2     b^2

where:

(+-a,0) are the x-intercepts; from (a,0) to (-a,0) is the transverse axis.
c^2 = a^2 + b^2

Now in your example, you need your a,b,c:

2c is the distance between the foci and clearly  c = 100.
2a is the difference between the distances.  In this case, you just multiply your 500 microseconds by 0.186 to get the value of 2a, then the value of a.

Finally, use the Pythagorean relation  c^2 = a^2 + b^2  to compute b^2 (you don't actually need b) and then you can write your equation.