Question

For the following exercises, sketch the graph of each conic. $$ r=\frac{4}{1+\cos \theta} $$

Short answer

The conic is a parabola opening to the left.

Step-by-step solution

Identify the Conic Type

The given equation is in the polar form \[r = \frac{e \, d}{1 + e \cos \theta}\]where \(e\) is the eccentricity, and \(d\) is the directrix. The equation given is \[r = \frac{4}{1 + \cos \theta}\]which compares directly to the standard form. Here, \(e = 1\), indicating that the conic is a parabola.

Determine the Direction of the Opening

In the polar conic form,\[r = \frac{ed}{1 + e \cos \theta}\]if the equation has \(+\cos \theta\), the conic opens to the left, which is the case here, given the equation\[r=\frac{4}{1 + \cos \theta}.\]

Graph Key Points of the Parabola

For specific values of \(\theta\), calculate \(r\) to plot points on the graph. When \(\theta = 0\),\[r = \frac{4}{1+1} = 2.\]When \(\theta = \frac{\pi}{2}\) or \(\theta = \frac{3\pi}{2}\),\[r = \frac{4}{1 + 0} = 4.\]

Sketch the Graph Using the Parabola Properties

Plot the points calculated in Step 3 and sketch the graph of the parabola opening to the left. The vertex of this parabola is at \(r = 2\) (when \(\theta = 0\)), confirming the direction.

Key concepts

Polar Coordinates

Polar coordinates are a different way of expressing points in a plane, distinct from the traditional Cartesian system. Instead of using horizontal and vertical distances, polar coordinates rely on the distance from a fixed point (usually the origin) and an angle from a fixed direction (usually the positive x-axis). In the polar coordinate system, a point is represented as \((r, \theta)\), where \(r\) is the radius or distance from the origin, and \(\theta\) is the angle in radians.

In our given equation \(r = \frac{4}{1 + \cos \theta}\), \(r\) changes based on the angle \(\theta\). This dynamic relationship allows us to plot the conic section in a polar coordinate graph. Understanding polar coordinates can help provide insights into the symmetry and positioning of conics like parabolas.

• \(r\) denotes the radial distance from the origin.
• \(\theta\) is the angle from the polar axis, typically the positive x-axis.
• Polar equations can sometimes simplify complex relationships by exploiting symmetry.

Eccentricity

Eccentricity is a crucial concept in the study of conic sections, which are curves obtained by slicing a cone with a plane. It determines the shape of the conic section. Eccentricity, denoted as \(e\), is defined in terms of how much a conic deviates from being circular.

- For a parabola, which is our focus, \(e = 1\).
- If \(e < 1\), the conic is an ellipse.
- If \(e > 1\), the conic is a hyperbola.
- When \(e = 0\), the shape is a perfect circle.

In the polar equation \(r = \frac{4}{1 + \cos \theta}\), since \(e = 1\), we identify this conic as a parabola. The value of eccentricity provides a direct link in understanding the curve's identity and assists in plotting and graphing it correctly.

Graph Sketching

Graph sketching is a step-by-step process that helps visualize a mathematical equation through plotting key points and understanding the behavior of the function. For polar coordinate graphs, especially involving conics, this includes considering how \(r\) changes with \(\theta\), and identifying notable positions such as the vertex or other symmetry points.

When sketching the graph of the equation \(r = \frac{4}{1 + \cos \theta}\), you:

• Calculate specific values of \(\theta\). For instance, when \(\theta = 0\), \(r = 2\).
• Consider symmetry: here, the parabola opens leftwards, as indicated by the positive sign in \(1 + \cos \theta\).
• Identify key points, such as when \(\theta = \frac{\pi}{2}\) or \(\theta = \frac{3\pi}{2}\), leading to \(r = 4\).

By plotting these points and understanding their connections, you can sketch the shape and opening direction of the resulting parabola.

Parabola Properties

Parabolas are a specific type of conic section with unique properties that distinguish them from ellipses and hyperbolas. Their distinctive U-shape curve is defined as all points equidistant from a single point, called the focus, and a line known as the directrix.

Key properties of a parabola include:

• The axis of symmetry: a vertical or horizontal line through the vertex, which divides the parabola into two mirrored halves.
• Vertex: the point where the parabola changes direction. In our polar example, it occurs at \(r = 2\), when \(\theta = 0\).
• Focus and directrix: these determine the 'tightness' or 'width' of the parabola.
• For the polar equation \(r = \frac{4}{1 + \cos \theta}\), the opening direction is to the left due to \(+\cos \theta\).

Understanding these properties helps in sketching and analyzing parabolas, ensuring that their graph is accurately represented.