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June 22nd, 2012, 01:14 AM   #1
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Rational Points on a circle

Ques-1 Let C be any circle with centre (0,). Prove that at most two rational points can be there on C. (A rational point is a point both of whose co-ordinates are rational numbers)

Ques 2- Prove that all the circles having their centres on a fixed line and passing through a fixed point also pass through another fixed point.(It seems obvious but how to prove it?)

Please help me on solving these questions.
Thanks!
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June 22nd, 2012, 03:09 AM   #2
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Q2. This is clearly untrue if the fixed point lies on the fixed line. If the fixed point doesn't lie on the line, the other fixed point is the reflection of the original point with respect to the line, since these points are equidistant from the centre, no matter where it lies on the line.
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June 22nd, 2012, 03:22 AM   #3
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Re: Rational Points on a circle

Q1.

Proof by contradiction (an outline; you'll need to fill in the details).

Assume there are three or more rational points.

Then at least two of those points must have different y values.

Choose that pair.

Either we have a diameter in which case the centre is the midpoint and so has rational co-ordinates,

Or we can construct the perpendicular bisector of the two points in question. It will go through the centre. And show that the y-coordinate is rational.

In both cases the y-coordinate of the centre is rational, contradicting the irrationality of root 2.
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June 22nd, 2012, 04:36 AM   #4
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Quote:
Originally Posted by ghostwalker
It will go through the centre. And show that the y-coordinate is rational.
How will it do that?
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June 22nd, 2012, 04:47 AM   #5
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Re:

Quote:
Originally Posted by skipjack
Quote:
Originally Posted by ghostwalker
It will go through the centre. And show that the y-coordinate is rational.
How will it do that?
The eqn of the perpendicular bisector can be written in the form

, where is the midpoint of your two selected points.

will all be rational values. Substituting x=0 will give the y co-ordinate of the centre, which will thus be rational.
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June 22nd, 2012, 08:50 AM   #6
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Re: Rational Points on a circle

But how does that prove that there can be only two rational points on the circle.?
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June 22nd, 2012, 09:15 AM   #7
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Re: Rational Points on a circle

Quote:
Originally Posted by guru123
But how does that prove that there can be only two rational points on the circle.?

This is a proof by contradiction, which in essence is:

We assume that there are more than two rational points on the circle.

From this we've shown that root 2 is rational.

But we know root 2 is not rational.

A contradiction.

Our assumption is therefore false, and there are not more than two rational points.
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June 22nd, 2012, 09:35 AM   #8
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Re: Rational Points on a circle

I tried to prove it other way too, can it be said correct?

As centre of the circle is (0,), any point on the circle can be represented by (0+rcos(x), +sin(x)), where r is radius of any circle and x is any angle between 0 to 2.
So for coordinates to be rational, cos(x) should have rational values, which only corresponds to x being 0,60,90,120,180.
when x=0, coordinates become (r,)
when x=60, coordinates become (r/2, (r*)/2))
when x=90, coordinates become (0,+r)
when x=120, coordinates become (-r/2, (r*)/2))
when x=180, coordinates become (-r,)

points have a chance of being rational only at x=60 and x=120, for other x points can never be rational. rational points can be zero as well but not more than 2.
therefore for any arbitrary circle with centre (0,), there are two maximum rational points.
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June 22nd, 2012, 09:45 AM   #9
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Re: Rational Points on a circle

Quote:
Originally Posted by guru123

So for coordinates to be rational, cos(x) should have rational values, which only corresponds to x being 0,60,90,120,180.
I refer you to:

http://planetmath.org/encyclopedia/R...AndCosine.html

I.e there are infinite number of angles for which both the sine and cosine are rational.
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June 22nd, 2012, 10:03 AM   #10
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Re: Rational Points on a circle

For some reaons I can't edit - nor find what the minimum number of posts is in order to do so: Ho, hum.

I was going to add, regardless of whether the sine and cosine are rational or not, you still have to take into account the radius of the circle, as that will effect the rationality of the points, for any given angle.
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