Question

In polar coordinates, the points \((r, \theta)\) and \((-r, \theta)\) are symmetric with respect to which of the following? (a) the polar axis (or \(x\) -axis) (b) the pole (or origin) (c) the line \(\theta=\frac{\pi}{2}\) (or \(y\) -axis) (d) the line \(\theta=\frac{\pi}{4}\) \((\) or \(y=x)\)

Short answer

The points \(r, \theta\) and \(-r, \theta\) are symmetric with respect to (b) the pole (or origin).

Step-by-step solution

- Understand polar coordinates

In polar coordinates, a point \(r, \theta\) is defined by a radius \(r\) from the origin and an angle \(\theta\) from the positive x-axis.

- Analyze given points

The points given are \(r, \theta\) and \(-r, \theta\). These represent two points, one at a distance \(r\) from the origin at angle \(\theta\) and the other at a distance \(-r\) at the same angle.

- Interpret the point with \(-r\)

The negative radius \(-r\) indicates the point is at the same angle \(\theta\) but in the opposite direction, across the origin. Essentially, \(-r\) means a reflection through the origin.

- Identify axis of symmetry

Points that are symmetric with respect to the origin satisfy the condition of being equidistant in opposite directions from the origin. This corresponds to symmetry around the pole (or origin).

- Select the correct option

Based on our analysis, the points \(r, \theta\) and \(-r, \theta\) are symmetric with respect to the origin. Thus, the correct answer is (b) the pole (or origin).

Key concepts

Polar Coordinates

Polar coordinates are a way of describing the position of a point in a plane. Instead of using the conventional Cartesian coordinates \((x, y)\), the polar coordinate system uses two values: the radius \(r\) and the angle \(\theta\). The radius \(r\) represents the distance from the point to the origin (center of the polar coordinate system), and the angle \(\theta\) represents the direction of the point from the origin, measured counter-clockwise from the positive x-axis.
To find the coordinates of a point in the polar system, imagine a line from the origin to the point. The distance of this line is \(r\) and the angle it makes with the positive x-axis is \(\theta\). For example, if \(r = 5\) and \(\theta = \frac{\pi}{4}\), then the point is 5 units away from the origin at a 45-degree angle from the positive x-axis.
Polar coordinates have many applications, especially in fields dealing with circular and rotational structures, as they make it easier to describe points in such systems compared to Cartesian coordinates. Examples include the study of waves, circular motions, and some areas of physics and engineering.

Symmetry in Polar Coordinates

Symmetry in polar coordinates helps us understand how points reflect across specific lines or points. Two points are symmetric with respect to some axis or point if they are mirror images according to that feature.
If two points share the same angle \(\theta\) but one has a positive radius \(r\) and the other a negative radius \(-r\), they are symmetric with respect to the origin. This means that they are positioned directly opposite to each other, equidistant from the origin. For instance, the points \( (r, \theta) \) and \((-r, \theta)\) reflect across the origin.
Let's visualize: imagine \(r\) as extending from the origin in one direction; \(-r\) would be the same length but in the opposite direction. Hence, the origin (or pole) acts as the mirror line, making \((r, \theta)\) and \((-r, \theta)\) reflective symmetrically across it.
This understanding is crucial for problems that involve symmetry in polar coordinates, like in analyzing patterns or determining points' relationships in physics and engineering contexts.

The Origin

In the polar coordinate system, the origin is referred to as the pole. It is the central point from which all other points are measured. The pole has coordinates \((0,0)\) in polar terms, which means the radius \(r\) is 0, irrespective of the angle \(\theta\).
When dealing with symmetry, the origin is often the point of reflection. This is because any point at \(r, \theta\) can be reflected to \(-r, \theta\) around the origin. Hence, the pole becomes the key reference point.
For instance, if we consider symmetry with respect to the origin, a point at \( (4, \frac{\pi}{3}) \) would reflect to \((-4, \frac{\pi}{3}) \), placing it directly opposite through the origin. This is a fundamental concept helping to determine symmetrical properties in many fields including physics, engineering, and graphical representations.
Understanding this helps to solve problems related to symmetry and reflections in exercises like the one presented, as well as in real-life applications such as designing symmetrical patterns or analyzing forces in a circular motion.