Problem 8 - Algebra and Trigonometry

Question

1–8 Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(0.4,\) vertex at \((2,0)\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{0.8}{1 - 0.4\cos(\theta)} \).

Step 1: Understanding the problem

We are asked to write a polar equation of an ellipse with a given eccentricity and vertex position. The focus of this ellipse is located at the origin, which is important for identifying if the ellipse is centered around the origin.

Step 2: Identifying eccentricity and semi-major axis

The eccentricity (\(e\)) of the ellipse is given as 0.4. The vertex, which lies on the semi-major axis, is at \((2,0)\). This means the semi-major axis \(a\) is 2 units long in the () direction.

Step 3: Using the formula for polar equations of conics

An ellipse in polar coordinates with focus at the origin is given by: \[ r = \frac{ed}{1 - e\cos(\theta)} \]where \(d\) is the semi-latus rectum. Since the vertex is on the semi-major axis at (2,0), this implies the ruather than calculating directly from the equation, we can use the relationship between eccentricity, semi-major axis, and semi-latus rectum: \(d = ae = 0.8\).

Step 4: Substituting values into the equation

With the determined values, \(a = 2\), \(e = 0.4\), and \(ed = 0.8\), we substitute into the polar equation:\[ r = \frac{2 \cdot 0.4}{1 - 0.4\cos(\theta)} = \frac{0.8}{1 - 0.4\cos(\theta)} \]

Key Concepts

Eccentricity

Eccentricity is a key parameter that describes the shape of a conic section. It is denoted by the symbol \(e\) and can take different values depending on the type of conic section:

• For an ellipse, the eccentricity is greater than 0 and less than 1. A smaller value of \(e\) indicates a shape closer to a perfect circle.
• If \(e = 1\), the conic is a parabola.
• For a hyperbola, \(e\) is greater than 1.

In this specific problem, we are dealing with an ellipse that has an eccentricity of 0.4. This value implies that the ellipse is more elongated than a circle but not as stretched as a parabola or hyperbola. The concept of eccentricity helps in understanding how "stretched" or "squashed" an ellipse is.
The equation \(r = \frac{de}{1 - e\cos(\theta)}\) directly incorporates eccentricity \(e\) to help identify the shape and orientation of conic sections in polar coordinates.

Ellipse

An ellipse is a smooth, closed, and symmetric curve, much like an elongated circle. It is defined by its two foci and its major and minor axes. The major axis is the longest diameter of the ellipse, and it goes through both foci. The point on the major axis closest to the center is called a vertex.
Ellipses have some interesting properties:

• All points on an ellipse are at unequal distances from its two foci.
• The sum of the two distances from any point on the ellipse to the foci is constant.

In polar coordinates, an ellipse with a focus at the origin can be represented by the equation \(r = \frac{ed}{1 - e\cos(\theta)}\), where \(d\) is the semi-latus rectum. For this equation, if the eccentricity \(e = 0.4\), and the vertex is at \((2,0)\), it suggests that the semi-major axis \(a\) is 2 and the distance from the origin to the vertex determines the length of \(a\) on that axis.

Polar Coordinates

Polar coordinates provide a convenient way to represent curves, especially conics like ellipses, in terms of a distance from a fixed point and an angle. Unlike Cartesian coordinates, which use \((x, y)\), polar coordinates make use of \((r, \theta)\), where:

• \(r\) is the radius or the distance from the origin to a point.
• \(\theta\) is the angle formed with the positive x-axis.

This system is particularly useful when dealing with problems where symmetry about a point rather than about an axis is present—like with conic sections focused at the origin.
In the context of the current exercise, the polar equation \(r = \frac{ed}{1 - e\cos(\theta)}\) is used to describe an ellipse centered at the origin. By substituting the known values of eccentricity \(e\) and semi-latus rectum \(d\), we yield the specific polar equation. This form allows us to express complex shapes in a simpler format, ideal for analyzing the geometry of conic sections.