Problem 52 - College Algebra

Question

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=-2 ; e=\frac{8}{3}\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{16/3}{1 - 8/3 \cos \theta} \).

Step 1: Understanding conics in polar form

In polar coordinates, a conic with a focus at the origin can be expressed as \( r = \frac{ed}{1 - e \cos \theta} \), where \(e\) is the eccentricity, \(d\) is the distance from the focus to the directrix, and \(\theta\) is the angle in radians.

Step 2: Identify the orientation of the directrix

The directrix is given as \(x = -2\). Since it is a vertical line to the left of the focus, the conic is oriented horizontally. This means that \(\cos \theta\) will be used in the equation for polar form.

Step 3: Identify the eccentricity and directrix

The eccentricity \(e\) is \(\frac{8}{3}\), and the directrix is \(x = -2\), so \(d = 2\) since the directrix is at a distance of 2 units from the y-axis (focus).

Step 4: Substitute values into the polar equation

Substitute \(e = \frac{8}{3}\) and \(d = 2\) into the standard form of the polar equation of a conic: \[ r = \frac{\frac{8}{3} \times 2}{1 - \frac{8}{3} \cos \theta} \].

Step 5: Simplify the equation

Simplify the expression: \[ r = \frac{\frac{16}{3}}{1 - \frac{8}{3} \cos \theta} \], which can also be written as \[ r = \frac{16/3}{1 - 8/3 \cos \theta} \].

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. These curves have unique shapes depending on the angle of the plane.

• Circle: Formed when the plane is perpendicular to the axis of the cone.
• Ellipse: Created when the plane cuts through both nappes, but at a lesser angle than a circle.
• Parabola: Occurs when the plane is parallel to one of the sides (generatrix) of the cone.
• Hyperbola: Forms when the plane cuts both nappes at an angle.

In polar coordinates, these conic sections can be represented with polar equations, often expressed as:\[ r = \frac{ed}{1 - e \cos \theta}\]where \(e\) is the eccentricity and \(d\) is the directrix.Understanding these basic forms helps predict and manipulate their equations for various orientations and directrixes.

Eccentricity

Eccentricity is a measure of how much a conic section deviates from being circular. Specifically, it's a numerical value that determines the shape of the conic section.

• Circle: Eccentricity \(e = 0\).
• Ellipse: Eccentricity \(0 < e < 1\).
• Parabola: Eccentricity \(e = 1\).
• Hyperbola: Eccentricity \(e > 1\).

In our exercise, with an eccentricity \(e = \frac{8}{3}\), the conic is a hyperbola. This is because the value \(\frac{8}{3}\) exceeds 1, indicating that the shape curves outward more steeply compared to an ellipse or circle.

Directrix

A directrix is a fixed line used in the definition of a conic section. For conics in polar coordinates, the focus is at the origin, and the position of the directrix influences the orientation and equation of the conic.In the given exercise, the directrix is situated at \(x = -2\). This indicates a vertical line, orienting the conic horizontally because the line lies parallel to the y-axis.The distance \(d\), or how far this directrix is from the origin, is 2 units (since it is at \(x = -2\)).Understanding the location and orientation of the directrix helps in correctly placing the conic and ensures the accurate use of cosines in the polar equation.

Polar Coordinates

Polar coordinates represent points in the plane using a distance from the origin and an angular coordinate. The system uses a radial distance \(r\) and an angle \(\theta\) to define a point's position.For conic sections in polar coordinates, the equations involve the eccentricity \(e\), directrix distance \(d\), and the angle \(\theta\). The conic's position can thus change visibly as the angle varies, reflecting its dynamic shape based on angular rotation rather than fixed Cartesian axes.Formulas like:\[ r = \frac{ed}{1 - e \cos \theta}\]highlight how the combination of polar angles and distances builds different conics. This system is particularly useful in problems requiring rotational symmetry or involving equations around a central point, like in our exercise with a focus at the origin.