find the max n min no of points, having rational coordinates, which can exist on the circle with centre (0, sqrroot2)????

If radius is not given then maximum is infinite (No minimum)

Let the point with rational co-ordinates be (x,y) . Let the radius of the circle be r.

 

x^2 + (y - sq. root 2 )^2 = r^2

i.e x^2 + y^2 + 2  - 2sqroot2 y = r^2

i.e x^2 / (2y) + y / 2 - sq.root 2 = r^2 / (2y)....eqn(1)

( I have divided both sides of the eqn. by y. So, I have taken the assumption that y is not equal to zero. I will later take the case when y = 0 )

 

Observe, the L.H.S of eqn.(1) consists of rational and irrational parts. The only irrational part is sqroot 2.  Therefore, r must also consist of rational & irrational parts with the irrational part being sq. root 2.

So, let r = a - sq.root 2 (where a > sq. root 2 )

Now, putting this value of r in eqn. (1), we have the following,

(x^2 + y^2)/ 2y - sq.root 2 = (a^2 + 2) / 2y - sq.root 2 a /y

 

Now, comparing the coefficients of rational & irrational parts,

x^2 + y^2  =  a^2 + 2 ........ eqn.(2)

a/y = 1 ........... eqn.(3)

Substituting a = y in eqn. (2), we have,

x^2 = 2 which means x is irrational which is a contradiction !! 

 

Now, when y = 0, from the first step of the problem, we have,

x^2 + 2 = r^2.

There can be such possible rational x.

So, for a given radius of the circle, the two possible points with rational co-rdinates can be ( sq. root { r^2 - 2 } , 0 ) & ( - sq. root { r^2 - 2 } , 0 ) provided

sq. root { r^2 - 2 } is rational.

 

I am pretty confident about my method. But plz point my mistakes, if any !!