Problem 45 - Precalculus

Question

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc}\text{Conic} & \text{Vertex or Vertices} \\\ \text{Parabola} & (1,-\pi / 2) \end{array}$$

Step-by-Step Solution

Answer

The polar form equation of the given parabola with its focus at the pole and vertex at (1,-\(\pi / 2\)) is \(r = \frac{1}{1 - cos(\theta)}\).

Step 1: Declare Given Variables and Constants

The given vertex is (1,-\(\pi / 2)\), and we know the focus is at the pole (0,0). In the general parabolic polar equation \(r = \frac{ed}{1 \pm ecos(\theta)}\), we know that \(e = 1\) (since this is a parabola) and \(d = 1\) (since the distance from the focus to the directrix is 1).

Step 2: Substitute Variables and Constants into Equation

Substitute the values of \(e\) and \(d\) into the general equation to simplify to: \(r = \frac{1}{1 \pm cos(\theta)}\).

Step 3: Determine Direction of Parabola

Since the vertex is in the fourth quadrant and the focus at the origin, the parabola points downwards or the direction is negative. This is interpreted as a negative cosine value, hence use the negative sign in the denominator.

Step 4: Write Down the Final Equation

After choosing the sign, the final equation of the parabola is \(r = \frac{1}{1 - cos(\theta)}\)

Key Concepts

Conic Sections

Conic sections are curves obtained by slicing a cone with a plane. These curves are fundamental in mathematics and include circles, ellipses, parabolas, and hyperbolas.
Understanding conic sections helps in the study of geometry, especially in different coordinate systems such as Cartesian and polar coordinates.

• **Circle**: All points are at a constant distance from a central point, forming a perfect loop.
• **Ellipse**: An elongated circle, or oval, with two focus points where the sum of distances from these points to each other on the curve is constant.
• **Parabola**: A symmetric curve where each point is equidistant from a single focus point and a directrix.
• **Hyperbola**: A set of all points where the difference of distances to two foci is constant.

Using conic sections in polar coordinates is particularly useful when the focus is at the origin, simplifying many equations and solutions.

Parabola

A parabola is a U-shaped curve that is the set of all points in a plane equidistant from a fixed point called the focus and a line called the directrix.
The standard quadratic form of a parabola is \(y=ax^2+bx+c\), but it can also be represented in polar coordinates when the focus is at the pole.
In polar form, parabolas have unique characteristics:

• **Vertex**: The point closest to or farthest from the pole, here given as (1, -\(\pi / 2\)).
• **Focus**: Located at the pole or origin (0, 0).
• **Equation**: \r = \frac{ed}{1 \pm e\cos(\theta)}\, where \(e = 1\) for parabolas.

These attributes help in painting a complete picture of a parabola's geometry and behavior in polar coordinates.

Focus at Pole

In polar coordinates, placing the focus at the pole (0,0) can greatly simplify the representation of conic sections, such as parabolas.
When the focus coincides with the pole, it results in a more straightforward equation, which is especially beneficial for calculation and graphing.
For a parabola, this removes any constant terms outside the fraction, and the variable terms guide how the shape will open or close around the pole.

• **Advantage**: Simplifies equations, reducing complexity.
• **Effect**: Leads to easier interpretations and increased accuracy in transformations and shifts.

This concept is pivotal in exercises involving polar equations, stemming from the cleaner routes they provide toward solutions.

Polar Coordinates

Polar coordinates, represented by \((r, \theta)\), provide a system where each point on a plane is determined by a distance \(r\) from a reference point and an angle \(\theta\) from a reference direction.
They offer a compelling alternative to Cartesian coordinates, especially useful for problems involving circularity or symmetry about one axis.
Key aspects include:

• **Radius (\(r\))**: Distance from the pole, which acts as the origin.
• **Angle (\(\theta\))**: Measured in radians from the positive x-axis.

Using polar coordinates to depict a parabola helps in visualizing curves that curve infinitely, akin to orbits, waves, and other circular phenomena.
They streamline the mathematical process when dealing with angles and rotations, making them ideal for a deeper understanding of geometry in dynamic systems.