Question

For the following exercises, consider the following polar equations of conics. Determine the eccentricity and identify the conic. $$ r=\frac{5}{-1+2 \sin \theta} $$

Short answer

Eccentricity is 2; the conic is a hyperbola.

Step-by-step solution

Identify the Conic Equation Form

The given equation is \( r = \frac{5}{-1 + 2 \sin \theta} \). This is in the standard form \( r = \frac{ed}{1 + e \sin \theta} \), but with a negative sign in the denominator, which indicates the conic is vertically oriented instead of horizontally.

Extract Parameters

From the equation \( r = \frac{5}{-1 + 2 \sin \theta} \), we see that the denominator resembles \(-1 + e \sin \theta\). Here, the parameter \( ed = 5 \), the constant term in the denominator is \(-1\), and the coefficient of \(\sin \theta\) is \(2\).

Calculate Eccentricity

Since the form is \( r = \frac{ed}{-1 + e \sin \theta} \), compare \(-1 + e \sin \theta\) with \(-1 + 2 \sin \theta\) to find the eccentricity: \( e = 2 \).

Identify the Type of Conic Based on Eccentricity

For conic sections, if \( e = 1 \), it is a parabola; if \( e < 1 \), it is an ellipse; and if \( e > 1 \), it is a hyperbola. Since \( e = 2 \), which is greater than 1, this conic is a hyperbola.

Key concepts

Eccentricity

Eccentricity is a crucial characteristic of conic sections, which helps to determine their shape and nature. It is represented by the symbol \( e \). In polar coordinates, the eccentricity effectively tells us how "stretched" or "compressed" the conic is.

• For a circle, the eccentricity \( e = 0 \).
• An ellipse has an eccentricity \( 0 < e < 1 \).
• A parabola's eccentricity is always \( e = 1 \).
• If the eccentricity \( e > 1 \), the conic section is a hyperbola.

In the provided exercise, the equation \( r = \frac{5}{-1 + 2 \sin \theta} \) was given. From the standard form \( r = \frac{ed}{-1 + e \sin \theta} \), we identify that \( e = 2 \). This value of eccentricity tells us that the conic is a hyperbola, according to its properties of eccentricity.

Conics

Conics refer to the different types of curves that can be generated by slicing a double-napped cone. These sections come in various forms, namely circles, ellipses, parabolas, and hyperbolas. Each form of the conic is determined by the angle and position of the intersecting plane relative to the cone.

• Circle: This is a special case of an ellipse where the plane is perpendicular to the axis of the cone, and all points are equidistant from the center (eccentricity \( e = 0 \)).
• Ellipse: Formed when the cutting plane is at an angle, but not parallel to the cone's side (\( 0 < e < 1 \)).
• Parabola: Occurs when the plane is parallel to an element of the cone (\( e = 1 \)).
• Hyperbola: Results when the plane intersects both nappes of the cone at an angle (\( e > 1 \)).

Understanding these conics is fundamental because they model numerous real-world phenomena, from the orbits of planets (ellipses) to the paths of comets (hyperbolas). In polar coordinates, identifying these conics can be simplified by examining their eccentricity values.

Hyperbola

A hyperbola is a type of conic section, which appears as two separate curves. It is formed when the plane intersects both nappes of the cone, creating two symmetric branches that mirror each other.

Here are some key features of hyperbolas:

• Two Branches: Unlike other conics, a hyperbola consists of two disconnected curves or branches.
• Eccentricity: For hyperbolas, the eccentricity \( e \) is always greater than 1. This means their curves are more "open" than those of ellipses.
• Asymptotes: Hyperbolas have asymptotes. These are straight lines that the branches approach but never touch. They determine the direction the hyperbola opens.

In the given exercise, with \( e = 2 \), we see that the curve described by the polar equation is indeed a hyperbola, confirming its property that \( e > 1 \). Understanding hyperbolas is valuable as they model many scenarios in physics and engineering, such as the paths particles can take in certain force fields.