Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Parabola} &(5, \pi)\end{array}$$

Step-by-Step Solution

Answer

The polar equation for the parabola with vertex at (5, \(\pi\)) and focus at the pole is given by \(r = \frac{5}{1 + \cos(\theta)}\).

Step 1: Identifying Eccentricity of Parabola

For all parabolas, the eccentricity \(e = 1\). This is a characteristic property of all parabolas.

Step 2: Determining the Distance of Vertex from Pole

This distance is given directly as 5 units. So, \(d = 5\). This is the maximum or minimum distance of the vertex from the pole.

Step 3: Formulating the Polar Form of Parabola

Now we substitute \(e = 1\) and \(d = 5\) in the equation of conics in polar coordinates. This simplifies to \(r = \frac{5}{1 \pm \cos(\theta)}\). \(\theta\) is the angle made by the line joining any point on the curve with the pole.

Step 4: Analysis of the Vertices

According to the vertex given, we can see it's in direction of the x-axis (\(\pi\)). Hence we use positive sign in denominator which gives our polar form as \(r = \frac{5}{1 + \cos(\theta)}\). This is our required polar equation of the parabola.

Key Concepts

Eccentricity of Parabola

The concept of eccentricity is critical in understanding the shapes of conic sections. With a distinct value for each type of conic section, it serves as a numerical indicator of a conic's shape. For a parabola, the eccentricity, denoted as \( e \), is always equal to 1. Unlike an ellipse or a hyperbola, which have eccentricities that differ from 1 (ellipses have \( e < 1 \) and hyperbolas have \( e > 1 \)), the parabola's eccentricity marks its unique property. In the polar coordinates system, this unvarying eccentricity simplifies the process of defining a parabola's equation. When students are asked to find the polar equation of a conic with its focus at the pole, recognizing that a parabola's eccentricity is 1 allows them to proceed confidently to the next steps of the problem-solving process.

Polar Coordinates

Polar coordinates provide a different approach to defining the position of a point in the plane, compared to the traditional Cartesian coordinates. Instead of using x and y coordinates, a point in the polar system is determined by the length of a straight line from the pole (or origin) to the point, which is the radial coordinate \( r \), and the angle \( \theta \) between this line and the polar axis (typically the positive x-axis).

This system is particularly useful when dealing with scenarios where symmetry around a central point, such as circular motion or patterns, makes it more practical than using rectangular coordinates. Polar equations are representations of curves in this coordinate system and can often reveal the nature of the curve more directly compared to Cartesian equations. The polar equation for a parabola with the focus at the pole is derived based on the relationship between the radial distance \( r \) and the angle \( \theta \), factoring in crucial parameters such as the eccentricity and the distance from the pole to the vertex of the parabola, as seen in the exercise above.

Vertex of a Parabola

The vertex of a parabola holds the key to understanding its structure and position. In the context of polar coordinates, the vertex is the point on the parabola that is closest to or furthest from the pole, depending on its orientation. As the polar equation of a parabola is centered on the focus at the pole, the vertex can be seen as the pivotal point from which the parabola's curve emanates.

In the exercise provided, the vertex is given by the coordinates (5, \( \pi \)), with 5 representing the distance from the pole (the radial coordinate) and \( \pi \) representing the angle of the vertex from the polar axis, which in this case, aligns with the positive x-axis. Decoding the vertex coordinates is an essential step in constructing the parabola's polar equation, as the vertex's radial distance essentially sets the parameter used in the equation, reflecting how wide or narrow the parabola will be.