Problem 41 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(C\) be any circle with centre \((0, \sqrt{2})\). Then, on the circle \(C\), there can be (A) at the most one rational point (B) at the most two rational points (C) at the most three rational points (D) none of these

Step-by-Step Solution

Answer

(D) None of these. No rational points on the circle with center \((0, \sqrt{2})\).

Step 1: Define what a Rational Point is

A rational point is a point on the Cartesian plane whose coordinates \((x, y)\) are both rational numbers. This means both \(x\) and \(y\) can be expressed as fractions of integers where the denominator is not zero.

Step 2: Identify the Circle Equation

The general equation of a circle with a center at \((0, \, \sqrt{2})\) is \((x - 0)^2 + (y - \sqrt{2})^2 = r^2\). This simplifies to \(x^2 + (y - \sqrt{2})^2 = r^2\).

Step 3: Investigate Rational Points on the Circle

For the circle to have rational points, \(y - \sqrt{2}\) must be rational, since \(x^2\) is already rational if \(x\) is rational. Let \(y = a + b\sqrt{2}\) where \(a\) and \(b\) are rational. For \(y\) itself to be rational, \(b\) must be zero (otherwise \(\sqrt{2}\), an irrational number, will be multiplied with a non-zero rational \(b\) yielding an irrational number). Therefore, \(y - \sqrt{2} = a\) must be zero, which implies no rational solutions unless \(b = 0\), making \(\sqrt{2}\) rational, which it is not. Thus, constructing a rational solution here is impossible.

Step 4: Evaluate the Possibility of Rational Points

Even if we attempt to solve for rational points rigorously, under our assumption, the circle equation with any rational solution results in a contradiction due to the irrational path involving \(\sqrt{2}\). There cannot be rational coordinates that make the distance from the origin a rational number without balancing terms involving \(\sqrt{2}\) hence confirming the non-existence of such points.

Key Concepts

Circle Equation

A circle equation in coordinate geometry represents all the points that form a circle around a specific center point. The general equation for a circle with center \( (h, k) \) and radius \( r \) is: \[(x - h)^2 + (y - k)^2 = r^2\] This formula makes it easy to identify whether a point \( (x, y) \) lies on the circle. If it satisfies the equation, the point is on the circle.
For the specific circle we discussed, with center at \( (0, \sqrt{2}) \), its equation becomes: \[x^2 + (y - \sqrt{2})^2 = r^2\]To find points on the circle, one needs to solve this equation for values of \( x \) and \( y \) that satisfy it. The parameters of this equation can be adjusted for different circles by changing \( h, k, \) or \( r \) to fit different scenarios or problem statements.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. In simpler terms, they are numbers that can be written as a fraction like \( \frac{a}{b} \).
Here are some examples of rational numbers:

• \( \frac{1}{2} \)
• \( \frac{-3}{4} \)
• \( 7 \) (as this can be written as \( \frac{7}{1} \))

Rational numbers are important in coordinate geometry because when we refer to a rational point, both the x and y coordinates must be rational numbers.
Finding rational points on a circle involves looking for coordinates \( (x, y) \), both of which need to be rational. For example, if the center of the circle or the radius itself involves an irrational number, such as \( \sqrt{2} \), like in our original problem, creating rational points becomes challenging.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. This allows for the description of geometric shapes in algebraic equations.
Coordinates are typically given in the form of \( (x, y) \) in a 2-dimensional plane, and using this system, we can study figures like lines, circles, and more complex shapes.
Understanding points on this plane is crucial as it allows solving problems related to finding distances, midpoints, and verifying particular criteria like points laying on a circle.

• The x-coordinate determines the horizontal positioning.
• The y-coordinate determines the vertical positioning.

The relationship between algebra and geometry through coordinates helps in solving complex problems. For example, by using the circle equation, one can analytically determine the set of points that lie on the circle.
This makes coordinate geometry not just about pictures, but about figuring out where those pictures are and how they interact using math. It’s a bridge between algebra and geometry that opens up new ways to solve spatial problems.