Question

For the following exercises, consider the following polar equations of conics. Determine the eccentricity and identify the conic. $$ r=\frac{5}{2+\sin \theta} $$

Short answer

Eccentricity is 2; the conic is a hyperbola.

Step-by-step solution

Identify the General Formula

The general formula for a polar equation of a conic is \( r = \frac{ed}{1 + e\sin\theta} \) or \( r = \frac{ed}{1 + e\cos\theta} \), where \( e \) is the eccentricity and determines the type of conic, and \( d \) is the distance from the directrix to the pole.

Compare Given Equation with General Formula

The given polar equation is \( r = \frac{5}{2 + \sin \theta} \). By comparing it with the general formula \( r = \frac{ed}{1 + e\sin\theta} \), we identify that \( ed = 5 \) and \( e = 2 \). Thus, \( e = 2 \) is the eccentricity.

Determine the Conic Type

The eccentricity \( e \) determines the type of conic:- If \( e = 1 \), it's a parabola.- If \( e < 1 \), it's an ellipse.- If \( e > 1 \), it's a hyperbola.Since \( e = 2 > 1 \), the conic is a hyperbola.

Key concepts

Eccentricity

Eccentricity is key when studying conic sections within polar equations. It defines how "stretched" or "flattened" a conic section looks.

• When the eccentricity ( \( e \) ) is less than one, the conic is an ellipse. This means the shape is more round, and each point on the ellipse is closer to a central focal point.
• If the eccentricity equals one, the conic becomes a parabola. Parabolas have a single focal point, and their arms extend infinitely in one direction.
• When the eccentricity is greater than one, the conic forms a hyperbola. This means the shape consists of two mirrored curves, each stretching away in opposite directions.

In our given exercise, the eccentricity is determined to be 2, which immediately tells us the conic section in question is not a circle or an ellipse, but instead another form, which we will describe next. This makes eccentricity a simple yet powerful indicator of the conic’s nature.

Conic Sections

Conic sections are formed by the intersection of a plane and a double-napped cone. Depending on the angle and the position of the plane, different shapes are created.

• Ellipses, including circles as a special type of ellipse where all eccentricities are zero, are formed when the plane intersects the cone at an angle less than that of the side of the cone.
• Parabolas arise when the plane is parallel to the side of the cone. They have a "U" shape because such an intersection opens symmetrically.
• Hyperbolas occur when the plane intersects both cones. This creates two separate curves pointing away from each other.

Each shape can be described using polar coordinates where the equations involve eccentricity and a directrix. By comparing the given equation with the standard formula, you can find parameters such as the eccentricity, helping to identify the specific conic section.

Hyperbola

Hyperbolas are one of the conic sections, noted for having two distinct branches that mirror each other. They appear when the eccentricity, \( e \), is greater than one, as shown in the exercise with \( e = 2 \).Key features of hyperbolas include:

• Asymptotes: Hyperbolas feature lines that the curve approaches but never touches. These asymptotes cross at the hyperbola's center.
• Vertices: The points on the hyperbola closest to the center.
• Foci: Each branch is drawn more closely to one of the two foci than to the other point, similar to an ellipse.

In polar coordinates, such curves are represented with equations reflecting the focal distance and direction. The exercise shows that by identifying the eccentricity as 2, the conic is determined as a hyperbola, characterized by its extensive spread and twin curves.