Circle with irrational centre

In summary, we are discussing the number of rational points on a circle with an irrational center. A rational point has both x and y as rational numbers, while an irrational point has at least one of x or y as irrational. By writing down the equations of the circle for two points and then adding a third point, we can prove that X and Y cannot be rational unless two of the rational points on the circle are identical or lie on a straight line. Therefore, the maximum number of rational points on a circle with an irrational center is two.
  • #1
vishal007win
79
0
How many rational points can be there on a circle which has an irrational centre?
(rational point is a point which have both x,y as rational numbers)

how to proceed??
answer is: atmost 2
 
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  • #2
You say a "rational point" has both x and y rational numbers. Is an "irrational point", then, a point that has either x or y or both irrational?
 
  • #3
I'd think that if a rational point has both x,y rational, an irrational point would be
a point that is not ratinal, so has at least one of x,y irratinal?

write down the equations of the circle for two points

[tex] (X - x_1)^2 + (Y - y_1)^2 = R^2 [/tex]
[tex] (X - x_2)^2 + (Y - y_2)^2 = R^2 [/tex]

You can eliminate R from them, and write them so X and Y become separated

[tex] 2 X (x_2 - x_1) + 2 Y (y_2 - y1) = x_2^2 + y_2^2 - x_1^2 - y_1^2 [/tex]

Note that all numbers, except for X and Y are rational. With 2 points it's still
possible to have X or Y irrational. Now add a third point

[tex] (X - x_3)^2 + (Y - y_3)^2 = R^2 [/tex]

and combine this equation with the one for the first point producing

[tex] 2 X (x_3 - x_1) + 2 Y (y_3 - y1) = x_3^2 + y_3^2 - x_1^2 - y_1^2 [/tex]

we now get 2 simultaneous linear equations for X and Y, and it is possible to prove that X and Y cannot be rational unless 2 of the rational points on the circle are identical, or the rational points on the circle lie on a straight line.
 

What is a "Circle with irrational centre"?

A "Circle with irrational centre" is a type of geometric shape where the centre point of the circle has an irrational number as its coordinates. This means that the centre point cannot be expressed as a fraction or decimal, and is instead an infinite non-repeating sequence of numbers.

How is the centre of a "Circle with irrational centre" determined?

The centre of a "Circle with irrational centre" is typically determined by taking the square root of an irrational number and using that as the x and y coordinates. For example, the centre of a circle with a radius of √2 would be (√2, √2).

What is the significance of a "Circle with irrational centre"?

"Circle with irrational centre" serves as a mathematical concept that helps us understand the properties of irrational numbers. It also demonstrates how geometric shapes can be constructed using abstract mathematical ideas.

What is the relationship between the radius and circumference of a "Circle with irrational centre"?

The radius and circumference of a "Circle with irrational centre" are related through the irrational number π. The circumference of the circle is equal to 2π times the radius, meaning that the circumference will also be an irrational number.

Can a "Circle with irrational centre" have a finite area?

No, a "Circle with irrational centre" will always have an infinite area. This is because the area of a circle is calculated using the formula A=πr^2, and since the radius is an irrational number, the area will also be irrational and therefore infinite.

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