Question

For the following exercises, find a polar equation of the conic with focus at the origin and eccentricity and directrix as given. $$ \text { Directrix: } x=4 ; e=\frac{1}{5} $$

Short answer

The polar equation of the conic is \( r = \frac{4}{5 - \cos(\theta)} \).

Step-by-step solution

Understanding Conic Sections in Polar Coordinates

In polar coordinates, a conic section with the focus at the origin is given by \( r = \frac{ed}{1 - e \cos(\theta)} \) for a horizontal directrix. The parameters are: \( e \), the eccentricity, and \( d \), the distance to the directrix.

Convert Directrix to Polar Form

The given directrix is \( x = 4 \). This can be seen as \( r \cos(\theta) = 4 \). Hence, the directrix in polar form is simply \( d = 4 \) for our substitution.

Use the Conic Section Formula

Substitute \( e = \frac{1}{5} \) and \( d = 4 \) into the formula for the conic section: \[r = \frac{\frac{1}{5} \times 4}{1 - \frac{1}{5} \cos(\theta)}\] Simplify this to find the polar equation.

Simplify the Equation

Carrying out the multiplication in the numerator, \( \frac{1}{5} \times 4 = \frac{4}{5} \). Thus, the polar equation becomes: \[r = \frac{\frac{4}{5}}{1 - \frac{1}{5} \cos(\theta)} = \frac{4}{5(1 - \frac{1}{5} \cos(\theta))} = \frac{4}{5 - \cos(\theta)}\]

Key concepts

Conic Sections

Conic sections are important geometric shapes that are formed by the intersection of a plane with a double-napped cone. These include circles, ellipses, hyperbolas, and parabolas. Each type is defined by its unique properties and equations.

In the context of polar coordinates, conic sections have equations that describe curves with respect to the origin, which serves as the focus. By setting parameters like eccentricity and directrix, we can determine the exact shape of the conic. The conic's focus is always at the origin in polar equations. When the directrix is horizontal or vertical, it simplifies understanding and computation.

Polar equations, such as the one used in the exercise, represent these conic sections by describing the radius or distance from any point on the curve to the origin based on the angle, \( \theta \).

Eccentricity

Eccentricity is a measure that defines the shape of a conic section. It is denoted by \( e \) and determines how "stretched" a conic is. It's a vital component in identifying the type of conic section you're working with:

• If \( e = 0 \), the conic is a circle.
• If \( 0 < e < 1 \), the conic is an ellipse.
• If \( e = 1 \), it is a parabola.
• If \( e > 1 \), the conic is a hyperbola.

In the given problem, the eccentricity \( e = \frac{1}{5} \) indicates an ellipse since it is between 0 and 1. This means the conic section will be oval-shaped, and all points on it maintain a constant ratio of distances to a particular line (the directrix) and the focus. The choice of \( e \) directly affects the steepness and shape of the ellipse, dictating how round or elongated it is.

Directrix

The directrix is a fixed line used in the geometric construction of a conic section together with the focus. The concept is essential for understanding conic sections because it helps define these curves mathematically.

In the polar coordinate system, the directrix helps to describe how we measure the distance to points on the conic. It's especially significant for ellipses and hyperbolas, where it provides a reference for calculations.

With the directrix provided as \( x = 4 \), we translate it into polar form. For horizontal or vertical directrices, converting to polar coordinates involves the trigonometric relationship \( r \cos(\theta) = d \) or \( r \sin(\theta) = d \), depending on orientation. This exercise uses a directrix of \( x = 4 \), resulting in a distance \( d = 4 \), simplifying our substitution into the conic section formula. By using this relationship, you find the specific position and layout of the ellipse.