‘A’ and ‘B’ are two fixed points in a field. A cyclist C moves such that ACB is always a right angle. In this context, which one of the following statements is correct?

1. Basics of Coordinate Geometry and Distance (basic)

Welcome to the world of Coordinate Geometry! At its heart, this subject is the bridge between Algebra and Geometry. It allows us to describe the position of any point in space using numbers, making it an essential tool for navigation, physics, and logical reasoning. Imagine a flat surface called the Cartesian Plane, which is divided by two perpendicular lines: the horizontal x-axis and the vertical y-axis. The point where they cross is the Origin (0,0).

To locate a point, we use an ordered pair (x, y). The 'x' tells us how far to move horizontally, and 'y' tells us how far to move vertically. This is very similar to how we use the New Cartesian Sign Convention in optics to measure distances from a reference point like the pole of a mirror Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.143. In geometry, distances to the right and upwards are typically positive, while distances to the left and downwards are negative.

The most fundamental operation we perform in this system is finding the distance between two points. If we have two points, A(x₁, y₁) and B(x₂, y₂), the straight-line distance (d) between them is calculated using the Distance Formula, which is derived from the Pythagorean theorem:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

Understanding distance is crucial because it defines geometric shapes. For instance, a circle is simply the collection (or locus) of all points that are at a constant distance from a fixed center. Even in simple observations, like looking at an image in a plane mirror, we see that the distance of the image behind the mirror is exactly equal to the distance of the object in front of it Science-Class VII, NCERT (Revised ed 2025), Light: Shadows and Reflections, p.161. This concept of "fixed distance" or "equidistance" is the foundation for solving complex analytical puzzles.

Term	Definition	Example
Abscissa	The x-coordinate (horizontal)	In (3, 5), 3 is the abscissa.
Ordinate	The y-coordinate (vertical)	In (3, 5), 5 is the ordinate.
Locus	A path traced by a point moving under a specific rule.	A circle is a locus of points equidistant from a center.

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.143; Science-Class VII, NCERT (Revised ed 2025), Light: Shadows and Reflections, p.161

2. Understanding the Locus of a Point (basic)

In logical reasoning and geometry, the term Locus (plural: loci) refers to the set of all points whose coordinates satisfy a given condition or rule. You can think of it as the path traced by a moving point that is 'constrained' by a specific law. For instance, when we study linear motion, the locus of an object moving along a straight track between two stations is simply a straight line Science-Class VII, Measurement of Time and Motion, p.116. Whether the motion is uniform or non-uniform, if the constraint is to stay on that track, the geometric shape of the path remains a line Science-Class VII, Measurement of Time and Motion, p.117.

The concept of 'locus' is not limited to physics; it appears in various disciplines to describe a 'place' or 'position' defined by a rule. In law, the term locus standi refers to the 'right to stand' or the specific position a person must hold to approach a court Indian Polity, Public Interest Litigation, p.309. Similarly, in economics, a linear demand curve is effectively the locus of all points that show the relationship between price and quantity demanded Microeconomics (NCERT class XII 2025 ed.), Theory of Consumer Behaviour, p.30.

To identify a locus, you must identify the rule. Some common examples include:

• Rule: The point is always at a fixed distance from a single center point. Locus: A circle.
• Rule: The point is always equidistant from two parallel lines. Locus: A third parallel line midway between them.
• Rule: The point (C) moves such that it always forms a 90° angle with two fixed points (A and B). Locus: A circle with AB as the diameter (based on Thales' Theorem).

Understanding the locus allows us to predict the trajectory of an object even if we cannot see the entire path at once.

Sources: Science-Class VII, Measurement of Time and Motion, p.116; Science-Class VII, Measurement of Time and Motion, p.117; Indian Polity, Public Interest Litigation, p.309; Microeconomics (NCERT class XII 2025 ed.), Theory of Consumer Behaviour, p.30

3. Properties of Circles: Chords and Diameters (basic)

To understand circular motion and spatial reasoning, we must first master the fundamental anatomy of a circle. A circle is defined as the set of all points in a plane that are at a fixed distance (the radius) from a central point. Any straight line segment connecting two points on the circle's boundary is called a chord. The most significant chord is the diameter, which passes directly through the center and is exactly twice the length of the radius. In practical applications, such as optics, we often refer to the diameter of a circular reflecting surface as its aperture Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.137. This measurement is crucial because it determines how much light or energy a surface can capture.

One of the most elegant properties of a circle is known as Thales' Theorem. It states that if you take the diameter of a circle as one side of a triangle and pick any other point on the circle's circumference as the third vertex, the angle formed at that third vertex will always be a right angle (90°). This means the diameter always subtends a 90° angle at any point on the boundary. In the context of analytical reasoning, if an object moves such that it always maintains a 90° angle relative to two fixed points, its path of motion must be a circle where those two points form the diameter.

In physical geography, we apply these circular properties to the Earth. Because the Earth is roughly spherical, the shortest distance between any two points lies along a Great Circle — a circle whose center coincides with the center of the Earth. While many circles can be drawn around the globe (like lines of latitude), the Equator is the only latitude that qualifies as a great circle because it represents the maximum diameter of the Earth Certificate Physical and Human Geography, GC Leong (Oxford University press 3rd ed.), The Earth's Crust, p.14. Understanding these geometric constraints helps us solve complex problems involving navigation, planetary motion, and even the shadows cast during eclipses.

Sources: Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.137; Certificate Physical and Human Geography, GC Leong (Oxford University press 3rd ed.), The Earth's Crust, p.14

4. Conic Sections: Circles vs. Ellipses (intermediate)

To master logical analytical reasoning, we must distinguish between shapes not just by how they look, but by the geometric rules (loci) that create them. A Circle is the most symmetrical conic section, defined as the set of all points at a constant distance from a single center. However, a crucial property often used in analytical problems is Thales' Theorem: if you have two fixed points A and B, any point C that moves such that the angle ∠ACB is always 90° will trace a perfect circle with AB as its diameter. This is a rigid geometric constraint. In geography, we see this perfect symmetry in Great Circles, such as the Equator, which bisects the Earth into two equal hemispheres Certificate Physical and Human Geography, The Earth's Crust, p.14.

In contrast, an Ellipse is an elongated curve defined by two fixed points called foci. While a circle has one radius, an ellipse has a major axis (the long way) and a minor axis (the short way). This shape is fundamental to understanding our universe; for example, the Earth travels around the Sun in an elliptical orbit rather than a perfect circle Science-Class VII, Earth, Moon, and the Sun, p.186. Because the distance between the Earth and the Sun varies, our orbital speed also changes—moving fastest at the perigee (closest point) and slowest at the apogee (farthest point) Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.257.

Understanding the difference is vital for logical deduction. If a problem describes a path constrained by a constant right angle relative to two points, the geometry dictates a circle. If the path involves varying distances from a central point or gravitational body where the 'center' is offset, you are likely dealing with an ellipse. As we see with Earth's latitudes, while the Equator is a great circle, other parallels of latitude form smaller circles as they move toward the poles, maintaining a circular shape but changing in scale Exploring Society: India and Beyond (NCERT Class VI), Locating Places on the Earth, p.14.

Feature	Circle	Ellipse
Key Constraint	Constant angle (90°) subtended from a diameter.	Sum of distances from two foci is constant.
Symmetry	Uniform curvature; all diameters are equal.	Varying curvature; has major and minor axes.
Real-world Example	The Equator (Great Circle).	Planetary orbits around the Sun.

Sources: Certificate Physical and Human Geography, GC Leong, The Earth's Crust, p.14; Science-Class VII, NCERT, Earth, Moon, and the Sun, p.186; Exploring Society: India and Beyond, NCERT Class VI, Locating Places on the Earth, p.14; Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.256-257

5. Exponential Curves and Non-Linear Motion (intermediate)

In logical reasoning and data interpretation, understanding how variables relate to one another is crucial. Most basic relationships are linear, meaning the rate of change is constant—if you double the input, the output doubles. However, the real world often moves in non-linear ways. A non-linear relationship is one where the change in the dependent variable does not move at a constant rate relative to the independent variable. As we see in economic data, like the Total Product (TP) curve, output might increase as labor is added, but the rate of that increase often fluctuates rather than staying a straight line Microeconomics (NCERT class XII 2025 ed.), Production and Costs, p.40. An exponential curve is a specific type of non-linear relationship where the rate of change is proportional to the current value. Imagine a population doubling every year; the growth starts slow but becomes incredibly steep very quickly. This differs from other non-linear shapes, such as a circle or an ellipse, which represent closed paths or periodic motion. In economics, we often plot these variables on a graph to visualize these behaviors, such as drawing demand curves where price and quantity have an inverse, often non-linear, relationship Microeconomics (NCERT class XII 2025 ed.), Theory of Consumer Behaviour, p.22. Identifying the geometry of motion is a key analytical skill. When we plot data—whether it's land under cultivation or the movement of a particle—the resulting shape tells us the underlying 'rule' of that system Economics, Class IX, The Story of Village Palampur, p.3. If the path curves back toward its origin, it might be circular; if it explodes upward at an increasing rate, it is likely exponential.

Type of Relation	Rate of Change	Visual Shape
Linear	Constant (Fixed)	Straight Line
Exponential	Compounding (Increasing)	Steep Upward Curve
Circular	Constrained by a Center	Closed Loop/Curve

Sources: Microeconomics (NCERT class XII 2025 ed.), Production and Costs, p.40; Microeconomics (NCERT class XII 2025 ed.), Theory of Consumer Behaviour, p.22; Economics, Class IX, The Story of Village Palampur, p.3

6. Thales' Theorem: Right Angles and Semicircles (exam-level)

At its heart, Thales' Theorem is a fundamental principle of geometry that describes a unique relationship between a circle and a right-angled triangle. It states that if you have a diameter of a circle, any point on the circumference of that circle will form a right angle (90°) with the two endpoints of that diameter. Conversely, the locus (the set of all possible positions) of a point $C$ that maintains a constant $90°$ angle relative to two fixed points $A$ and $B$ is exactly the circle that has the segment $AB$ as its diameter.

This concept is vital for logical reasoning because it allows us to predict the path of motion based on angular constraints. Imagine a scenario where a person is moving, but they are physically or logically constrained to always view two fixed landmarks at a perpendicular angle. Just as we use geometry to map the path of light through a prism Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.165 or measure the angle of deviation Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.166, we can determine that this person's path must be circular. They aren't moving in an ellipse or a straight line; the geometry of the right angle forces them onto the arc of a circle.

Understanding these geometric paths helps us conceptualize motion and shape in broader contexts, such as the sphericity of the Earth. Just as circumnavigation proved the Earth's shape Certificate Physical and Human Geography, GC Leong, The Earth's Crust, p.4, Thales' Theorem proves that certain angular relationships are only possible on specific geometric curves.

Constraint	Resulting Locus (Path)
Fixed distance from a single point	Circle (Radius)
Fixed 90° angle between two points	Circle (Diameter)
Refraction through a prism	Angular deviation path

Sources: Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.165; Science, Class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.166; Certificate Physical and Human Geography, GC Leong, The Earth's Crust, p.4

7. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental properties of triangles and circles, this question perfectly demonstrates the application of the locus of a point. In coordinate geometry, when we say points A and B are fixed and point C moves under a specific constraint—in this case, maintaining a constant 90-degree angle—we are essentially looking for the geometric path that satisfies this condition. You should immediately recognize this as a classic application of Thales' Theorem, which you just studied. This theorem states that the angle subtended by a diameter at any point on the circumference is always a right angle.

To arrive at the correct answer, visualize the line segment AB as the diameter of a circle. If the cyclist C moves such that ∠ACB is always 90°, then C must always lie on the boundary of that circle. Reasoning through the geometry, if you were to pick any point on a circular perimeter and connect it to the ends of the diameter, you would always form a right-angled triangle. Therefore, the only way for the cyclist to maintain this constant perpendicularity relative to the fixed points A and B is to follow the curvature of a circle. This makes (B) The path followed by the cyclist is a circle the only logical conclusion.

UPSC often uses distractors like "ellipse" or "exponential curve" to catch students who are guessing based on the complexity of the terms. While an ellipse is a closed curve, its definition relies on the sum of distances from two foci being constant, not a constant angle. The "not possible" option is a classic psychological trap designed to make you doubt the mathematical principles you've learned. By sticking to the standard results found in Circle Geometry, you can confidently identify that this motion is not only possible but mathematically certain to be circular.