AMU 2026 — Mathematics PYQ

1. Understanding Polar Coordinates

In a polar coordinate system, any point in a two-dimensional plane is represented by a pair of coordinates $(r,\theta )$, where:

• $r$ represents the radial distance (the straight-line distance from the central origin point, known as the pole).

• $\theta$ represents the angular coordinate or polar angle measured counter-clockwise from the initial positive horizontal axis.

2. Definition of a Circle Centred at the Pole

By definition, a circle is the geometric locus of all points in a plane that maintain a fixed, constant distance (the radius) from a fixed central point.

If the fixed central point is located exactly at the pole $(0,0)$, then the distance from the pole to any point $(r,\theta )$ lying on the circumference of that circle must always equal the radius $a$:

$$\text{Distance from the pole}=r=a$$

Since the distance from the center is entirely independent of the direction or angle $\theta$, the variable $\theta$ can take any value from $0$ to $2\pi$ radians ($180^{\circ }$ to $360^{\circ }$). Thus, the equation simplifies strictly to:

$$r=a$$

3. Analysis of Other Options (For Reference)

• Option (b) $r=2a\cos \theta$: This represents a circle of radius $a$ passing through the pole, with its centre lying on the positive polar axis at the coordinates $(a,0)$.

• Option (c) $r=2a\sin \theta$: This represents a circle of radius $a$ passing through the pole, with its centre lying on the vertical axis at the coordinates $\left(a,\frac{\pi }{2}\right)$.

Conclusion

The simplest polar equation representing a circular profile centered precisely at the origin (pole) with a radius value of $a$ is given by $r=a$.

This directly matches option (a).