Problem 50 - 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: \(y=\frac{2}{5} ; e=\frac{7}{2}\)

Step-by-Step Solution

Answer

The polar equation is \(r = \frac{14/5}{1 + \frac{7}{2}\sin(\theta)}.\)

Step 1: Understand the Type of Conic

Conics are defined by their eccentricity, denoted as \(e\). If \(e = 1\), the conic is a parabola. If \(e < 1\), it is an ellipse. If \(e > 1\), it is a hyperbola. Given \(e = \frac{7}{2} = 3.5 > 1\), this will be a hyperbola.

Step 2: Identify the Orientation

The given directrix is \(y = \frac{2}{5}\), which is a horizontal line. This implies that the conic is vertically oriented. This is important for setting up the polar equation.

Step 3: Recall the Polar Equation of a Conic

The general polar equation for a conic section with the focus at the origin is \(r = \frac{ed}{1 + e\sin(\theta)}\) when the directrix is horizontal and \(r = \frac{ed}{1 + e\cos(\theta)}\) when the directrix is vertical. Here since the directrix is \(y = \frac{2}{5}\), we use \(\sin(\theta)\).

Step 4: Determine the Distance to Directrix

The directrix \(y = \frac{2}{5}\) represents distance \(d\) from the focus (the origin) to the directrix. In this case, \(d = \frac{2}{5}\).

Step 5: Substitute Values into the Equation

Using the polar equation form \(r = \frac{ed}{1 + e\sin(\theta)}\) for a vertical directrix with \(e = \frac{7}{2}\) and \(d = \frac{2}{5}\), we substitute to get:\[r = \frac{\frac{7}{2} \cdot \frac{2}{5}}{1 + \frac{7}{2}\sin(\theta)} = \frac{\frac{7 \times 2}{5 \times 2}}{1 + \frac{7}{2}\sin(\theta)} = \frac{\frac{14}{5}}{1 + \frac{7}{2}\sin(\theta)}.\]

Step 6: Simplify the Polar Equation

After performing the calculations, the polar equation is:\[r = \frac{14/5}{1 + \frac{7}{2}\sin(\theta)}.\]

Key Concepts

Understanding Conic Sections

Conic sections are the curves obtained by slicing a double cone with a plane. Depending on the angle and position of the slice, you get different curves known as conics. Here are the main types of conic sections you might encounter:

• Circle: A slice perpendicular to the cone's axis produces a circle.
• Ellipse: An angled slice hitting both halves without going through the base forms an ellipse.
• Parabola: Slicing parallel to the slant of the cone gives a parabola.
• Hyperbola: Cutting at a steep angle that goes through both cones creates a hyperbola.

Conic sections have intriguing properties and play significant roles in mathematics and science. They are useful in fields ranging from astronomy to engineering. Understanding these basic shapes helps lay the groundwork for more complex topics, like polar equations of conics.

Eccentricity Demystified

Eccentricity describes how "stretched" or "squashed" a conic section appears. It's represented by the letter \(e\). Depending on its value, eccentricity helps identify the type of conic section:

• If \(e = 0\), the conic is a circle, typically seen as a special case of an ellipse.
• If \(0 < e < 1\), the conic is an ellipse.
• If \(e = 1\), the conic is a parabola.
• If \(e > 1\), the conic is a hyperbola.

In the given problem, \(e = \frac{7}{2} = 3.5\), indicating a hyperbola. Eccentricity is crucial in determining the geometric shape and orientation of the conic sections. The eccentricity governs how wide or narrow a hyperbola appears, impacting its polar equation representation.

The Role of the Directrix

The directrix is a crucial line related to conic sections, serving as one of the two key reference lines. The properties of conic sections can be expressed in terms of their distances from a particular focus and this directrix.

• A directrix is a straight line that helps define a conic section in conjunction with a focus point.
• For ellipses and hyperbolas, a directrix can provide a frame of reference to understand how the curve bends or stretches.
• The distance from any point on a conic to its focus and directrix always holds a specific ratio, dictated by the eccentricity \(e\).

In polar equations, the location of the directrix, whether vertical or horizontal, influences the trigonometric function used (\(\sin\) for horizontal, \(\cos\) for vertical). In this exercise, the directrix \(y = \frac{2}{5}\) positioned horizontally requires the use of \(\sin(\theta)\) in the equation, affecting the form of the resulting hyperbola.

Decoding Hyperbolas

Hyperbolas are fascinating conic sections that form when a plane intersects both nappes (the two cones) in steep angles. They consist of two separate curves known as branches.

• A hyperbola has two symmetrical parts called branches that open either vertically or horizontally depending on its orientation.
• The hyperbola's axes are defined by the transverse and conjugate axes, where the transverse axis intersects both branches.
• It's described by its eccentricity \(e > 1\), and grows wider as \(e\) increases.

In polar coordinates, hyperbolas centered at the origin take the form \(r = \frac{ed}{1 + e\sin(\theta)}\) or \(r = \frac{ed}{1 + e\cos(\theta)}\) depending on the alignment of the directrix. This polar representation helps in visualizing the surface and understanding its geometry based on eccentricity and orientation.