Question

For the following exercises, determine the eccentricity and identify the conic. Sketch the conic. $$ r=\frac{7}{5-5 \cos \theta} $$

Short answer

The conic is a parabola with an eccentricity of 1.

Step-by-step solution

Identify the Conic Equation Format

The given equation is in polar form: \[ r = \frac{7}{5 - 5\cos\theta} \]This is similar to the standard form \( r = \frac{ed}{1 + e\cos\theta} \) or \( r = \frac{ed}{1 - e\cos\theta} \), where \( e \) is the eccentricity and \( ed \) is a constant.

Re-write the Equation for Easier Comparison

Rearrange the given equation:\[ r = \frac{7}{5(1 - \cos\theta)} \]This helps in identifying the denominator as a factor of the standard form \( r = \frac{ed}{1 - e\cos\theta} \).

Determine the Eccentricity

Compare the equation \( r = \frac{7}{5(1 - \cos\theta)} \) to \( r = \frac{ed}{1 - e\cos\theta} \).From this comparison, we find:- \( ed = 7 \)- \( 5e = 5 \)Solving for \( e \), we get:\[ e = 1 \]

Identify the Conic Section

Since the eccentricity \( e = 1 \), the conic section is a parabola. A parabola is defined by having an eccentricity equal to 1.

Sketch the Conic

A conic with an equation of the form \( r = \frac{ed}{1-e\cos\theta} \) with \( e=1 \) is a parabola with its directrix perpendicular to the polar axis.- Plot points using various values of \( \theta \) to illustrate how the graph behaves as a parabola opening to the right.- Note that at \( \theta = 0 \), the distance \( r \) from the origin to the conic is maximum, corresponding to the vertex of the parabola. The directrix is a horizontal line to the left of the parabola.

Key concepts

Eccentricity

The eccentricity of a conic section is a measure of how much it deviates from being circular. It is a key parameter in defining the shape of the conic section:

• An eccentricity ( \( e \) ) of 0 represents a circle.
• If 0 < \( e \) < 1, the conic is an ellipse.
• For \( e = 1 \) , we have a parabola.
• If \( e > 1 \) , the conic is a hyperbola.

In the given equation \[ r = \frac{7}{5(1 - \cos\theta)} \], we find that the eccentricity ( \( e \) ) is equal to 1. This is determined by comparing the equation to the standard polar form, where \( 5e \) equals the coefficient of \( \cos\theta \) , which is 5. Solving for \( e \) gives:\[ e = \frac{5}{5} = 1 \]Therefore, this conic section is a parabola, characterized by its eccentricity of 1.
Understanding eccentricity is crucial as it not only identifies the conic but also gives insight into its geometric properties.

Polar Coordinates

Polar coordinates provide a different way of representing points in the plane compared to Cartesian coordinates. Instead of using an \( x \) and \( y \) coordinate, polar coordinates use:

• \( r \) : the radial distance from the origin (center).
• \( \theta \) : the angular coordinate, measured in radians.

In polar equations like \( r = \frac{7}{5 - 5\cos\theta} \), the variable \( \theta \) represents the angle of the line extending from the origin to the point, rotated from the positive x-axis. The variable \( r \) tells how far along that line the point lies from the origin.
This system is particularly beneficial for expressing conics like the one in our exercise, because it naturally aligns with the radial symmetry of these shapes. Plotting points with various values of \( \theta \) can help sketch the shape and identify key features such as vertices and axes of symmetry.
Conics in polar coordinates often involve \( \cos\theta \) or \( \sin\theta \) terms, making them compatible with equations of circles, ellipses, and parabolas.

Parabola

A parabola is a unique type of conic section. In the context of our problem, it is defined by its specific polar equation form: \( r = \frac{ed}{1 - e\cos\theta} \) , where the eccentricity ( \( e \) ) is equal to 1. This equation describes a parabola opening along the direction determined by \( \theta \) .Key features of a parabola include:

• The vertex, which is the closest point of the parabola to the origin in a polar graph.
• The directrix, a line perpendicular to the polar axis that is used to define all points of the parabola.
• The focus, which, in this case, is at the origin due to the nature of polar coordinates.

In our exercise, at \( \theta = 0 \) , the distance \( r \) reaches its maximum, marking the vertex of the parabola. As \( \theta \) varies, the curve extends, forming the typical U-shape of a parabola, symmetric about the direction line. Understanding these properties allows the accurate sketching and interpretation of parabolas in polar coordinates.