Problem 40 - Precalculus

Question

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc}\text{Conic} & \text{Eccentricity} & \text{Directrix} \\\ \text{Parabola} & e=1 & y=-4 \end{array}$$

Step-by-Step Solution

Answer

The polar equation of the conic is \(r = \frac{4}{1 + \cos(\theta)}\).

Step 1: Identification

Identify the type of conic section, the eccentricity, and the directrix. In this case, it is a parabola with \(e=1\) and directrix \(y=-4\), respectively.

Step 2: Calculate distance

Calculate the distance \(d\) between the focus and the directrix. As the directrix is \(y = -4\), the distance \(d\) to the origin is 4.

Step 3: Substitute values into the formula

Substitute the values into the formula for polar coordinates with a focus at the pole. For a parabola, this equation is \(r = \frac{d}{1 + e\cos(\theta)}\). Substituting \(d=4\) and \(e=1\) gives \(r = \frac{4}{1 + \cos(\theta)}\).

Key Concepts

Conic Sections

A conic section is a curve obtained as the intersection of a cone (a three-dimensional figure) with a plane. The angle between the plane and the cone's axis determines the type of conic section: circle, ellipse, parabola, or hyperbola.

• Circle - Occurs when the cutting plane is perpendicular to the cone's axis.
• Ellipse - Forms when the cutting plane is angled less steeply than the cone's side but does not cross the base.
• Parabola - Results when the plane is parallel to a single generatrix of the cone.
• Hyperbola - Occurs when the plane crosses both halves of the cone, creating a mirror image.

Each conic section can be represented mathematically and has unique properties. In the given exercise, the task was to find the polar equation of a parabola, a unique conic section without a second intersecting loop.

Eccentricity

Eccentricity is a number that describes the shape of a conic section. It is denoted by the letter 'e' and can provide a great deal of information about the conic's configuration.

• Circle: Has an eccentricity of 0, which signifies a perfect round shape.
• Ellipse: The eccentricity is between 0 and 1, indicating an elongated circular shape.
• Parabola: This is defined by an eccentricity equal to 1, revealing that it is open-ended and extends infinitely.
• Hyperbola: Has an eccentricity greater than 1, showing it consists of two diverging curves.

Understanding eccentricity is key to differentiating between the conic sections and finding their equations. For the parabola in the textbook problem, the given eccentricity is 1, indicating its shape and guide to the general equation used in polar coordinates.

Polar Coordinates

Polar coordinates offer an alternative to Cartesian (x, y) coordinates for describing the position of points in a plane, especially when dealing with circular or rotational symmetry. A point is defined by the distance from a reference point called the pole (equivalent to the origin in Cartesian coordinates) and an angle from a reference direction.

• The distance is usually denoted by r (radius)
• The angle is denoted by θ (theta), measured in radians or degrees from the positive x-axis.

With polar coordinates, the equation of a conic section can be expressed elegantly, as seen in this exercise. To solve for a conic with the focus at the pole, the formula involves the eccentricity and the polar coordinates, leading to an insightful way of representing curves that would be more complex in Cartesian terms. The resultant equation is tailored to the unique properties of the particular conic—parabolas in this case—and lends itself well to analyzing and graphing their behavior.