Problem 25 - Algebra and Trigonometry

Question

A polar equation of a conic is given. (a) Show that the conic is a hyperbola, and sketch its graph. (b) Find the vertices anddirectrix, and indicate them on the graph. (c) Find the center of the hyperbola, and sketch the asymptotes. $$ r=\frac{8}{1+2 \cos \theta} $$

Step-by-Step Solution

Answer

The conic is a hyperbola with vertices at \( r=\frac{16}{3} \), directrix at \( x = -8 \), and center at (0,0).

Step 1: Identify the Type of Conic Section

We start with the given polar equation \( r = \frac{8}{1 + 2\cos\theta} \). A polar equation of the form \( r = \frac{ed}{1 + e\cos\theta} \) represents a type of conic section. If \( |e| > 1 \), the conic is a hyperbola. Here, \( e = 2 \), which is greater than 1, indicating that the conic is a hyperbola.

Step 2: Sketch the Hyperbola

To sketch the hyperbola, first note that the conic has its eccentricity \( e = 2 \), with the focus located at the origin. This suggests the hyperbola opens in a direction based on the positive component, which is the positive \( x \)-axis because of \( \cos\theta \). Draw two branches diverging outward along the x-axis.

Step 3: Find Vertices and Directrix

For polar equations of conics, the vertex is at \( r = \frac{ed}{e+1}\). Substitute \( e = 2 \) and \( d = 8 \) to get vertices at \( r = \frac{16}{3} \). The directrix is a line perpendicular to the axis of the hyperbola at distance \( d = 8 \) from the focus (origin) on the negative \( x \)-axis.

Step 4: Determine the Center of the Hyperbola

In polar coordinates, the center refers to the original focus from which measurements are taken, which is at the pole \((0,0)\).

Step 5: Sketch the Asymptotes

The asymptotes of a hyperbola do not appear directly in polar coordinates, but can be inferred by understanding the angle of divergence from the pole. Asymptotes in Cartesian coordinates tend to infinity where the conic branches open up, corresponding to the lines the hyperbola approaches as \( r \to \infty \).

Key Concepts

Conic sections

Conic sections are curves formed by the intersection of a plane and a double-napped cone. This intersection produces different shapes based on the angle of the plane. Conic sections include:

• Circle: The plane cuts perpendicular to the axis of the cone.
• Ellipse: The plane cuts at an angle, but not through the base.
• Parabola: The plane is parallel to a generating line of the cone.
• Hyperbola: The plane cuts through both naps of the cone.

Each type of conic section has a unique equation and geometrical properties. Hyperbolas have an eccentricity greater than 1 and consist of two separate branches opening in opposite directions. In this exercise, we dealt with a hyperbola as identified by the eccentricity value obtained from the polar equation.

Polar equations

Polar equations express mathematical relations using polar coordinates, which differ from the typical Cartesian coordinates. In polar form, a point's position is described by:

• Radial distance (\(r\)): The distance from the origin to the point.
• Angular coordinate (\(\theta\)): The angle formed by the positive x-axis and the line connecting the origin to the point.

For conic sections, polar equations can take the form \( r = \frac{ed}{1 + e\cos\theta} \) or \( r = \frac{ed}{1 + e\sin\theta} \). These represent different scenarios depending on the angle indicator, either \(\cos\) or \(\sin\), which affects the orientation of the conic. The parameter \(e\) is the eccentricity, while \(d\) represents a fixed distance related to the conic's geometry. Polar coordinates are particularly useful for representing curves that exhibit radial symmetry.

Eccentricity of hyperbola

Eccentricity is a measure of how much a conic section deviates from being circular. It is denoted by the symbol \(e\). For different conic sections:

• Circle: \(e = 0\)
• Ellipse: \(0 < e < 1\)
• Parabola: \(e = 1\)
• Hyperbola: \(e > 1\)

In the polar equation of a hyperbola, the eccentricity determines the shape and opening of the hyperbola. A higher eccentricity means a greater distance between the branches of the hyperbola. This exercise's hyperbola has an eccentricity \(e = 2\), confirming its hyperbolic nature. The hyperbola's branches open based on the value's position relative to the positive axial directions in the polar coordinate system.