Find how many points on a circle have an integer distance from other points

[songoku]

Homework Statement

Please see below

Relevant Equations

$(x-a)^2+(y-b)^2=r^2$

Distance between 2 points

Distance between point (-4, 5) and point on circle:
$$d=\sqrt{(x+4)^2+(y-5)^2}$$
$$=\sqrt{x^2+8x+16+y^2-10y+25}$$

Then substitute $y^2$ from equation of circle:
$$d=\sqrt{x^2+8x+16-x^2+4x-6y+12-10y+25}$$
$$=\sqrt{12x-16y+53}$$

After this, I need to try the points one by one to check whether the distance is an integer and also whether the point is located on the circle? I am pretty sure there should be a more sane method to do this

Thanks

 

[hutchphd]

I would start by changing coordinates to put the origin at circle center. Then there are only a few right triangles with integer sides. Actually I'll bet you can show it must be an even number,which is all you need for this.
More later if necessary .

 

[songoku]

Thank you very much hutchphd

 

[Hall]

(Inspired from @hutchphd ’s reply):

Can you workout the center of the circle? We would get it as (2,-3). And radius?

Take any point on the circle, draw a triangle joining the center, the point (-4,5) and the arbitrary point on the circle (x,y). We know the distance between the point on circle and the center is $r$ (please workout it out) and the distance between the center and (-4,5) is 10 units. With resepct to the angle made at the center in this triangle, can you write out the cosine rule?

 

[hutchphd]

With resepct to the angle made at the center in this triangle, can you write out the cosine rule

Or just "complete the square" to find the original radius.
I seem to be having trouble getting any of the prescribed answers although my symmetry argument seems correct. But I have been wrong before!

 

[PeroK]

First, a change of coordinates to $\bar x = x - 2, \ \bar y = y + 3$ is a good idea.

Then use the symmetry to move the point $P$ to somewhere simpler, keeping the same relationship with the circle.

Then count.

However, as the circle itself has an integer radius ...

 

[hutchphd]

It does? Jeez I must have screwed up. I wish I could do arithmetic.

 

[SammyS]

After this, I need to try the points one by one to check whether the distance is an integer and also whether the point is located on the circle? I am pretty sure there should be a more sane method to do this

Thanks

No, you only need to check the distance for at most 2 points on the circle. Those being the point nearest to $(-4,~5)$ and the one farthest from $(-4,~5)$.

How many integers fall between those two extremes?

 

[hutchphd]

No, you only need to check the distance for at most 2 points on the circle. Those being the point nearest to (−4, 5) and the one farthest from (rs by symmetry−4, 5).

Except I believe the points (almost) always come in equidistant pairs...

 

[SammyS]

Except I believe the points (almost) always come in equidistant pairs...

I don't claim that's the entire solution.

Let's leave something for song-o to figure out.

(Added later, just so you don't think I'm time traveling)

Yes, I also get one of the listed answers.

 

[hutchphd]

Did you get one of the multiple listed answers (we can do that much)?

 

[PeroK]

Did you get one of the multiple listed answers (we can do that much)?

Yes.

 

[hutchphd]

I will look forward to the complete answer at the appropriate time ...!

EDIT DUH!: When you complete the square the quadratic terms are always positive. I wish I could do arithmetic. Thanks.

 

[songoku]

I got the answer using @hutchphd hint in post#2 but the method I use is not as efficient as @SammyS in post#8

But sorry I can't continue the hint given by @PeroK in post#6

 

[PeroK]

But sorry I can't continue the hint given by @PeroK in post#6

By symmetry, the answer must be an even number.

 

[songoku]

By symmetry, the answer must be an even number.

If all the options were even numbers, how to move point P to obtain the answer?

Thanks

 

[PeroK]

If all the options were even numbers, how to move point P to obtain the answer?

Rotate the system to put $P$ on the $x$-axis.

 

[songoku]

Thank you very much Hall, hutchphd, SammyS, PeroK

 

[Prof B]

To calculate the answer you need the radius of the circle and the distance between P and the center of the circle. After that the coordinates are irrelevant.

 

[robphy]

Since the problem appears solved, here are some possibly interesting visualizations of the problem.
(Try the "interesting views" at the bottom.)

https://www.desmos.com/calculator/cf30ycwkta

There's a hint of a related physics problem: interference from two coherent point sources.