Problem 14 - College Algebra

Question

Identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r(1-\cos \theta)=3 $$

Step-by-Step Solution

Answer

The conic is a parabola with eccentricity 1 and directrix at distance 3 from the origin.

Step 1: Rewriting the Equation

Given the equation for the conic in polar form: \[ r(1 - \cos \theta) = 3 \]We can rewrite it as:\[ r = \frac{3}{1 - \cos \theta} \] This equation is in the form of \[ r = \frac{e \, d}{1 - e \cos \theta} \] where the conic section is a conic with the focus at the origin.

Step 2: Identifying Eccentricity

The standard equation for a conic with focus at the origin and directrix perpendicular to the polar axis is\[ r = \frac{e \, d}{1 - e \cos \theta} \]In our equation\[ r = \frac{3}{1 - \cos \theta} \]we identify that the eccentricity \(e = 1\) because it matches exactly the term \(1 - e \cos \theta\). This means it is a parabola, as the eccentricity \(e = 1\) defines a parabolic shape.

Step 3: Finding the Directrix

Since the eccentricity \(e = 1\), the general form \[ r = \frac{e \, d}{1 - e \cos \theta} \] simplifies to\[ r = \frac{d}{1 - \cos \theta} \]where \(d = 3\). This means the directrix is a line perpendicular to the polar axis at a distance \(d = 3\) from the focus.

Key Concepts

Polar Coordinates

Polar coordinates are a way of expressing points in a plane, particularly useful for dealing with problems that have circular or radial symmetry. Unlike Cartesian coordinates, which use X and Y coordinates to signify a point on a plane, polar coordinates describe a point using two values: the radius ( ) and the angle ( heta).

• The radius ( ) describes how far a point is from the origin.
• The angle ( heta) indicates the direction of the radius, typically measured in radians or degrees from a reference direction, usually the positive X-axis.

Polar coordinates are particularly useful in scenarios where circular motion or rotation is involved. For example, planets orbiting a star or an object rotating around a central point. It simplifies these problems by naturally aligning with their circular or spiral paths.
The given equation \[ r(1 - \cos \theta) = 3 \] was expressed in polar form. We were able to rewrite it to better analyze the characteristics of the contained conic section.

Eccentricity

Eccentricity ( e ) is a key value that helps define the traits of conic sections such as parabolas, ellipses, and hyperbolas. This dimensionless number indicates how much a conic section deviates from being a perfect circle. In essence, eccentricity determines the shape of the conic section.

• If \(e = 0\), the conic section is a circle.
• If \(0 < e < 1\), it forms an ellipse.
• If \(e = 1\), a parabola.
• If \(e > 1\), a hyperbola.

In the exercise, the equation was rewritten as \[ r = \frac{3}{1 - \cos \theta} \] Here, you can equate it to the standard form \[ r = \frac{e \, d}{1 - e \cos \theta}, \] which helps identify that \(e = 1\). This implies that the conic section described is a parabola, characterized by its property of having no closed loop like ellipses or hyperbolas.

Directrix

The directrix is a crucial line associated with a conic section in polar coordinates. It helps us understand and construct the geometry of the curve. For a conic section with a focus, the directrix is a line such that each point on the conic maintains a constant ratio of the distance to the focus and to the directrix.
In this exercise, given the conic's polar equation form \[ r = \frac{d}{1 - \cos \theta} \] and knowing the eccentricity \( e = 1 \), it simplifies our equation to \[ r = \frac{d}{1 - \cos \theta} \] with \(d = 3\). This indicates that the directrix is positioned at a distance of 3 units from the focus, perpendicular to the polar axis.

• In a parabola, this directrix serves as a symmetry line ensuring all points are equidistant from both the focus and directrix, establishing the unique shape of a parabola in the coordinate system.