Problem 10 - College Algebra

Question

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r=\frac{16}{4+3 \cos \theta} $$

Step-by-Step Solution

Answer

It's an ellipse with eccentricity \( e = \frac{3}{4} \) and directrix \( d = \frac{64}{3} \).

Step 1: Identifying the Conic Type

The given equation is of the form \( r = \frac{ed}{1 + e\cos\theta} \). For this form, it represents a conic with a focus at the origin. Since the equation is \( r = \frac{16}{4+3\cos\theta} \), we can rewrite it as \( r = \frac{16}{4(1 + \frac{3}{4}\cos\theta)} \). Therefore, \( e = \frac{3}{4} \). The conic type is determined by the eccentricity \( e \). Since \( e < 1 \), the conic is an ellipse.

Step 2: Finding Eccentricity

The eccentricity \( e \) of the conic is given by the coefficient of \( \cos\theta \) in the denominator of the equation divided by the constant term. Therefore, the eccentricity is \( e = \frac{3}{4} \).

Step 3: Determining the Directrix

The directrix \( d \) of the conic is determined using the formula \( d = \frac{a}{e} \), where \( a \) is the numerator of the fraction on the right-hand side of the equation. Since \( a = 16 \) and \( e = \frac{3}{4} \), we have \( d = \frac{16}{1} = \frac{16}{\frac{3}{4}} \). Simplifying gives \( d = \frac{16}{1} \times \frac{4}{3} = \frac{64}{3} \). So, the directrix is \( \frac{64}{3} \).

Key Concepts

Ellipse

An ellipse is an elongated, closed curve. You can think of it as a stretched out circle. Unlike the perfect symmetry of a circle, an ellipse has two points inside known as foci. These points are key to its unique shape. The classic property of an ellipse is that the sum of the distances from any point on the ellipse to the foci is constant. This property creates that perfect, stretched out shape.

In our exercise, the equation given is characteristic of an ellipse, specifically because the eccentricity ( defined later) is less than 1. This indicates that the path traced by the points fulfilling the equation forms an ellipse instead of a hyperbola or parabola, where the eccentricity would be greater than or equal to 1, respectively.

When dealing with ellipses in polar coordinates, like in the given equation, recognizing the formula is essential to identifying the conic section involved. Let's move on to understand what eccentricity means.

Eccentricity

Eccentricity is a crucial value used to distinguish different types of conic sections. It measures how much an ellipse deviates from being a perfect circle. The formula to find the eccentricity, represented by the letter 'e', is dependent on the conic equation.

In simple terms:

• For circles, which are a special type of ellipse, the eccentricity is 0.
• For ellipses, the eccentricity lies between 0 and 1, not including 1.
• For parabolas, the eccentricity is exactly 1.
• For hyperbolas, the eccentricity is greater than 1.

In our specific exercise, we calculated the eccentricity to be \( e = \frac{3}{4} \), which is less than 1, confirming the curve is an ellipse. The smaller the eccentricity, the closer the shape is to a circle. Thus, with \( e = \frac{3}{4} \), the ellipse is moderately elongated.

Understanding eccentricity helps in visualizing and graphing ellipses more accurately, as it directly influences their shape.

Directrix

The directrix is a straight line used alongside one of the foci to construct a conic section. It's not part of the ellipse itself but is a reference line that helps define the conic's shape in relation to its eccentricity.

The definition is mathematical: a conic section is formed by the locus of a point such that its distance from a fixed point (focus) is proportional to its distance from a fixed line (directrix). This proportionality is shown in our equation by the eccentricity.

To find the directrix of an ellipse, particularly in the context of polar coordinates, you use the formula \( d = \frac{a}{e} \). Here, 'a' is the numerator of the given polar equation. In our case, it's 16. Dividing by the eccentricity \( \frac{3}{4} \), we found the directrix to be \( \frac{64}{3} \).

Knowing the directrix is important when constructing or analyzing a conic as it tells us how the conic section aligns compared to the reference line, aiding in precision during graphing or when solving related problems.