Question

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=6 $$

Short answer

The curve is a circle with eccentricity 0.

Step-by-step solution

Understanding the Polar Equation

The given polar equation is \( r=6 \). This equation describes a set of points in the polar coordinate system where each point is exactly 6 units away from the origin. This forms a circle centered at the origin.

Identifying the Curve

Since \( r=6 \) results in a constant radius from the origin, it describes a circle, not a conic section like an ellipse, parabola, or hyperbola, which would require a changing distance from the focus depending on the angle.

Eccentricity of Circle

A circle's eccentricity is 0 because it is a special case of an ellipse where the two foci coincide at the center, leading to no variation in distance at any point on the curve.

Sketching the Graph

To sketch this graph, draw a circle centered at the origin of the polar coordinate system with radius 6. Every point on this circle is at a distance 6 units from the center.

Key concepts

Conic Sections

Conic sections are a fascinating set of curves that are produced when a plane intersects with a double cone. These curves, including circles, ellipses, parabolas, and hyperbolas, have unique properties and characteristics. Each type of conic is defined based on the angle and location of that intersection with the cone.

While a circle is considered a conic section, it is often treated separately because of its constant radius and symmetrical properties. Unlike ellipses, which are elongated circles, or parabolas and hyperbolas, which have more complex shapes, a circle's shape is perfectly round. This is because it is formed when the plane intersects the cone at a perpendicular angle relative to its axis.

In polar coordinates, a circle is described by a simple equation like the one we have: \( r = k \), where 'k' represents the circle's radius. This simplicity distinguishes it from other conics that require more elaborate expressions to describe their varying radii.

Eccentricity

Eccentricity is a measure that helps to define the shape of a conic section. It tells us how much the shape deviates from being a perfect circle. This attribute is essential as it provides insight into the conic's geometric properties.

For different conics:

• A circle's eccentricity is 0, indicating no deviation because every point is equidistant from the center.
• An ellipse has an eccentricity between 0 and 1, where 0 indicates a perfect circle, and values closer to 1 mean a more stretched shape.
• A parabola has an eccentricity of exactly 1. It opens infinitely in one direction.
• A hyperbola's eccentricity is greater than 1, meaning it has two branches that diverge away from each other.

Understanding eccentricity helps in identifying and differentiating conic sections in both theoretical and practical applications.

Graph Sketching

Graph sketching, especially in a polar coordinate system, is an excellent skill for understanding mathematical concepts visually. It involves plotting points based on their angle and distance from the origin, referred to as the pole in polar coordinates.

When sketching the graph of the equation \( r = 6 \) in polar coordinates, you are essentially drawing a circle:

• Locate the origin, which serves as the center of the graph.
• Using the radius provided, in this case, 6, draw the circle so that every point on its circumference is exactly 6 units away from the origin.
• Ensure uniformity in the distance to maintain a perfect circle shape.

This approach of representing and analyzing mathematical expressions through graphs builds a better intuitive understanding of the geometry involved.