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        riddles >> putnam exam (pure math) >> Re: Geometry question.
        
(Message started by: flamingdragon on Dec 12^th, 2006, 6:40pm)

Title: Re: Geometry question.
Post by flamingdragon on Dec 12^th, 2006, 6:40pm

Is there a point in the middle of the circle?

Title: Re: Geometry question.
Post by Sameer on Dec 12^th, 2006, 6:50pm

can you edit the post for better title? locus means the set of all poiints P that will satisfy your conditions... so it will be some sort of curve equation

as for e.g.

consider a circle around origin... consider point x on (-r,0) co-ordinate...

Consider a chord a angle t w.r.t positive x axis intersecting circle at P(x,y)
Then height = y = lsint
distance of the projection of chort on x-axis from origin = (r-lcost)
Using pythagoras you get l = 2rcost

Also x^2 + y^2 = l^2 so the midpoint P1 would be l/2 in lenght giving x^2+y^2=l^2/4=r^2cos^t

where -pi/2 < t < pi/2 that would be the locus for one set of points.. i forget the name of the curve tho... now as you rotate x around the circle you will get this curve rotated by that much angle.... i hope i had my math right...

Title: Re: Geometry question. (about Locuses (word?))
Post by towr on Dec 13^th, 2006, 1:09am

I'm not entirely familiar with the terminology; what is a chord, is it a line-piece between x and a point on the circle?
In that case, I think, you'd get a circle half the size of Y, with its center halfway between x and the center of Y. That also collapses all three seperate cases into one btw..

Title: Re: Geometry question. (about Locuses (word?))
Post by Grimbal on Dec 13^th, 2006, 3:17am

Yep, I vote for the circle. But not necessarily half the size of Y. It has a diameter that goes from the point x to the center of Y.
The trick is to understand that x, the midpoint of the chord and the center of Y form a right angle.

And I think the plural of locus is loci, and it means "locations".

Title: Re: Geometry question. (about Locuses (word?))
Post by towr on Dec 13^th, 2006, 5:32am

on 12/13/06 at 03:17:38, Grimbal wrote:

The trick is to understand that x, the midpoint of the cord and the center of Y form a right angle.

?!
I must be fundamentally mistaken about what a chord is then. I'd have though the chord could be tangent to the circle (If x is outside it). But if what you say is true that's impossible.

Maybe someone could draw it. Or maybe I could look it up in mathworld.. Whichever takes the least effort of my part..

Title: Re: Geometry question. (about Locuses (word?))
Post by SMQ on Dec 13^th, 2006, 5:38am

My understanding is that a chord is a line segment connecting any two points on a circle; e.g. a diameter is a chord which additionally passes through the center of the circle.

--SMQ

Title: Re: Geometry question. (about Locuses (word?))
Post by towr on Dec 13^th, 2006, 5:57am

on 12/13/06 at 05:38:27, SMQ wrote:

My understanding is that a chord is a line segment connecting any two points on a circle; e.g. a diameter is a chord which additionally passes through the center of the circle.

So we have a line through x that intersects the circle, and the linepiece in the circle is the chord?

Well, that completely changes things.
Except when x is on the circle, in which case you'll get a circle half the size that touches x and the center of Y.

Title: Re: Geometry question. (about Locuses (word?))
Post by SMQ on Dec 13^th, 2006, 6:18am

on 12/13/06 at 05:57:50, towr wrote:

[...] when x is on the circle [...] you'll get a circle [...] that touches x and the center of Y.

Actually, I believe that's what you'll get for any X (for X outside Y you only get the portion of the circle within Y).

--SMQ

Title: Re: Geometry question. (about Locuses (word?))
Post by Sameer on Dec 13^th, 2006, 10:42am

on 12/13/06 at 05:57:50, towr wrote:

How come I am getting the equation to be x^2+y^2=r^2cost??? This according to me is not a circle...

Title: Re: Geometry question. (about Locuses (word?))
Post by towr on Dec 13^th, 2006, 11:20am

on 12/13/06 at 10:42:44, Sameer wrote:

How come I am getting the equation to be x^2+y^2=r^2cost??? This according to me is not a circle...

I'm not sure how you got that

We have l = 2r cos(t),
taking half, and putting it back into the parametric equation (l cos(t) - r, l sin(t))
We get
(r cos²(t) - r, r cos(t) sin(t))
->
(r/2 [2cos²(t) - 1) - r/2, r/2 [2 cos(t) sin(t)])
using double angle formulas that's
(r/2 cos(2t) - r/2, r/2 sin(2t))
or
r/2( cos(2t), sin(2t)) + (-r/2, 0)
Which looks perfectly circular to me.

Title: Re: Geometry question. (about Locuses (word?))
Post by Barukh on Dec 14^th, 2006, 4:10am

koonie, consider the following purely geometric argument.

Title: Re: Geometry question. (about Locuses (word?))
Post by Icarus on Dec 14^th, 2006, 5:42am

To be more explanatory (don't look until you've tried to figure it out yourself):

[hideb]The perpendicular to a chord passing through the center of the circle always intersects the chord at its midpoint. Let Y be the circle, with center O, and let L be the circle having X and O as endpoints of a diameter. Because XO is a diameter, for any other point M on L, the angle XMO is right. Therefore if we extend XM to a chord of the original circle (assuming that XM intersects the original circle), we must have that M is the midpoint of that chord, or conversely, that the midpoint of the chord lies on L. We can even say this in the unique case when M = X if we consider the line XM to be the tangent to L at X. Every line passing through X intersects L in such a point M, so every chord of Y which passes through X must have its midpoint on L, and every point on L which lies within Y is the midpoint of such a chord.

I.e. Regardless of where X lies, the locus of points satisfying the condition is the portion of L that lies within Y.[/hideb]

___________________________________________________

This brings up the curious question in the case of X outside Y of the points of the circle L that are outside Y. By analogy, they should in some sense be the midpoint of a "chord" of Y passing though X. How can one define this "chord"?

Title: Re: Geometry question. (about Locuses (word?))
Post by SMQ on Dec 14^th, 2006, 8:23am

on 12/14/06 at 05:42:51, Icarus wrote:

By analogy, they should in some sense be the midpoint of a "chord" of Y passing though X. How can one define this "chord"?

Consider the parametric interpretation of the problem where a chord of the unit circle lies on a line passing through point X = (a,b). The equation for the line can be given in terms of a parameter t as (x,y) = (a+ut, b+vt) where (u,v) is a unit vector, i.e. u²+v²=1.

The endpoints of the chord are the intersections of this line with the unit circle: x²+y²=1. Substituting gives (u²+v²)t² + 2(au+bv)t + (a²+b²-1) = 0, and since (u,v) is a unit vector this further reduces to t² + 2(au + bv)t + (a²+b²-1) = 0.

Solving this equation can give complex results for t when (a,b) lies outside the unit circle, but this situation could be envisioned as an imaginary z axis extending above and below the x-y plane. Now chords outside of the unit circle are ~~"diagonal"~~ perpendicular to the x-y plane, but their midpoint is still in the x-y plane; specifically the point given by t = -(au+bv).

Alternatively by taking only the real portion of the resulting values of t, the entire chord can effectively be projected into the x-y plane. I'll try to attach a diagram of the resulting structure if I get a chance.

[edit]Er, strike that, the resulting chords are perpendicular, not diagonal, and so the projection into the x-y plane is just a point.[/edit]

--SMQ

Title: Re: Geometry question. (about Locuses (word?))
Post by Icarus on Dec 16^th, 2006, 9:29am

This is a way of seeing what it ought to be, but it doesn't solve the problem by itself. To solve the problem, you have to show that every such midpoint must lie on your circle (there are infinitely many of them, so just plotting a few doesn't cut it), and you also have to show that every point on the circle is the midpoint of chord through X.

If you cannot show these two things, then there is always the possibility that you failed to notice other midpoints that don't lie on the circle, or that some points on the circle were not the midpoints of chords.

Barukh's diagram shows the way to prove both.