Question

What is the polar equation of a circle of radius \(|a|\) centered at the origin?

Short answer

Answer: The polar equation of a circle with radius \(|a|\) centered at the origin is \(r^2 = a^2\), or alternatively \(r = |a|\) if the radius is positive.

Step-by-step solution

Write down the Cartesian equation of the circle centered at the origin

The equation of a circle centered at the origin with radius \(|a|\) in Cartesian coordinates is given by: \( x^2 + y^2 = a^2 \)

Convert the Cartesian coordinates to polar coordinates

To convert the Cartesian coordinates to polar coordinates, we use the transformations: \(x = r \cos θ\) \(y = r \sin θ\)

Substitute polar coordinates into the Cartesian equation

Replace \(x\) and \(y\) in the Cartesian equation with their polar coordinate expressions: \((r \cos θ)^2 + (r \sin θ)^2 = a^2\)

Simplify the equation

Expand and simplify the equation: \(r^2 \cos^2 θ + r^2 \sin^2 θ = a^2\) Factor out \(r^2\) from the left-hand side: \(r^2 (\cos^2 θ + \sin^2 θ) = a^2\) Since \(\cos^2 θ + \sin^2 θ = 1\), our equation becomes: \(r^2 = a^2\)

Write down the polar equation of the circle

From the simplified equation, the polar equation of the circle centered at the origin with radius \(|a|\) is given by: \(r^2 = a^2\) In general, if the circle's radius is positive, we can write the polar equation as \(r = |a|\).

Key concepts

Cartesian Coordinates

Cartesian coordinates are a well-established way to locate points on a plane. They rely on a pair of numerical values named 'x' and 'y,' which represent a point's horizontal and vertical distances from a fixed origin point. Imagine a grid where you can plot a point by moving 'x' units right (or left) and 'y' units up (or down). This method is quite intuitive because it reflects our usual way of navigating through space.
- The Cartesian system captures the exact position of a point relative to two intersecting, perpendicular lines called axes (the x-axis and the y-axis).- The origin is typically placed at the center, denoted by the point (0,0).

A widely used application of Cartesian coordinates is to define geometric shapes, like circles. In the Cartesian system, a circle with radius \(a\) and centered at the origin is given by the equation \(x^2 + y^2 = a^2\). This equation simply means that no matter where you are on the circle, your distance from the center is constant, equal to \|a\|. This foundational idea sets the stage for converting to other coordinate systems, like polar coordinates.

Polar Equation

Polar coordinates provide a different, often advantageous way to represent points on a plane, especially useful for dealing with circular shapes. Instead of 'x' and 'y', polar coordinates use 'r' (the radius) and 'θ' (the angle). These coordinates specify a point by how far it is from a central origin and the angle it makes with a reference direction, typically the positive x-axis.
- To convert Cartesian coordinates to polar coordinates, you use the relationships \(x = r \cos θ\) and \(y = r \sin θ\), where \(θ\) is measured in radians.- When you apply these transformations to a circle, you quickly notice the simplicity of polar coordinates.

For example, the polar equation of a circle with radius \|a\|, centered at the origin, is simply \(r = |a|\). This highlights a key benefit of the polar system: it often reduces complex Cartesian equations into simpler, more natural forms that relate directly to the shape's intrinsic properties.

Circle Equation

The equation of a circle is a mathematical representation that helps describe its properties and shape on a plane. There are different forms this equation can take, depending on the coordinate system used.
- In Cartesian coordinates, the equation for a circle centered at the origin with a radius of \|a|\ is \(x^2 + y^2 = a^2\). This equation means that all the points \( (x, y) \) that satisfy this statement lie on the circumference of the circle.

This approach is quite flexible, allowing for the representation of circles centered at any point on a plane by adjusting the coordinates of the center.- In polar coordinates, however, the equation simplifies astonishingly to \(r^2 = a^2\) and then to \(r = |a|\). This simplicity underscores the advantages of polar coordinates for circular shapes, where the main concern is the distance from a central point, making the radius the key factor.
Knowing both forms of circle equations allows for versatile problem-solving and provides deep insight into the nature of circles in both Cartesian and Polar systems.