Let C be any circle with centre (0, √2). Prove that at the most two rational points can be there on

Q: Wait, why do we assume $r^2$ is the same for both points? Isn't $r$ just a constant?

Yes, $r$ is the radius, which is constant for the whole circle. By subtracting the equations, we eliminate the unknown $r^2$ entirely, which is a powerful technique in coordinate geometry to remove variables we do not know.

Q: Why does $y_1 - y_2$ have to be zero? Can't the irrational part just cancel out?

Great question! If $y_1 - y_2$ were non-zero, the right side would be irrational. But the left side is a difference of squares of rational numbers, which is rational. A rational number cannot equal an irrational number. Thus, the coefficient of $\sqrt{2}$ must be zero.

Q: Does this proof work if the center was $(0, 0)$?

No! If the center were $(0, 0)$, the equation would be $x^2 + y^2 = r^2$. There is no irrational term to force the points onto a line. In fact, a circle centered at the origin can have infinitely many rational points (e.g., Pythagorean triples).

Question

Let $C$ be any circle with centre $(0, \sqrt{2})$. Prove that at the most two rational points can be there on $C$. (A rational point is a point both of whose coordinates are rational numbers.)

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The standard equation of a circle is $(x-h)^2 + (y-k)^2 = r^2$. Substituting our center $(0, \sqrt{2})$, we obtain the equation:

$$x^2 + (y - \sqrt{2})^2 = r^2$$

Expanding the term $(y - \sqrt{2})^2$ yields $y^2 - 2\sqrt{2}y + 2$. Thus, the full equation of our circle is:

$$x^2 + y^2 - 2\sqrt{2}y + 2 = r^2$$

The term $-2\sqrt{2}y$ acts as the irrational intruder that disrupts the symmetry of the rational plane.

### The Clash of Rationality and Irrationality

Suppose there are two distinct rational points on this circle, $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$. Since they lie on the circle, they must satisfy the following equations:

$$x_1^2 + y_1^2 - 2\sqrt{2}y_1 + 2 = r^2$$

$$x_2^2 + y_2^2 - 2\sqrt{2}y_2 + 2 = r^2$$

To eliminate the unknown radius $r^2$, we subtract the second equation from the first. This yields:

$$(x_1^2 + y_1^2) - (x_2^2 + y_2^2) - 2\sqrt{2}(y_1 - y_2) = 0$$

Rearranging this to isolate the irrational component, we get:

$$(x_1^2 + y_1^2) - (x_2^2 + y_2^2) = 2\sqrt{2}(y_1 - y_2)$$

### The Revelation

Observe the left-hand side of the equation. Since $x_1, y_1, x_2,$ and $y_2$ are all rational, their squares and sums are necessarily rational.

For the equation to hold, the right-hand side must also be rational. Because $2\sqrt{2}$ is irrational, the only way the product $2\sqrt{2}(y_1 - y_2)$ can be rational is if the rational factor is zero.

Therefore, we must have $y_1 - y_2 = 0$, which implies $y_1 = y_2$. This proves that any two rational points on this circle must share the exact same $y$-coordinate.

### The Final Symmetry

Substituting $y_1 = y_2$ back into our subtracted equation, the $y$ terms vanish completely. We are left with:

$$x_1^2 - x_2^2 = 0 \quad \Rightarrow \quad x_1^2 = x_2^2$$

This implies $x_1 = \pm x_2$. If $x_1 = x_2$, the points are identical; if $x_1 = -x_2$, we have two distinct points symmetric about the $y$-axis.

A circle can intersect a horizontal line at most at two points. Since all rational points must lie on this specific horizontal line, it is geometrically impossible to have a third rational point.

We have successfully navigated the irrationality and proven that the circle contains **at most two rational points**.

Animated Solution for Mathematics - Circles: Let $C$ be any circle with centre $(0, 2)$ . Prove that at the most two rational points can be there on $C$ . (A rational point is a point both of whose coordinates are rational numbers.)

Visualized Solution

Visualizing the Circle and Center

Defining the Circle Equation

Expanding the Equation

Assuming Two Rational Points

Eliminating the Radius

The Rationality Constraint

Deducing the y-coordinates

Solving for x-coordinates

Final Conclusion

The Sigma Insight: Standard and General Equation of a Circle

Analyzing the Setup

The Clash of Rationality and Irrationality

The Revelation

The Final Symmetry

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