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Chapter 8: Problem 50

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(y=\frac{2}{5} ; e=\frac{7}{2}\)

Short Answer

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

Step by step solution

01

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.
02

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.
03

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 \(\sin(\theta)\).
04

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, \(d = \frac{2}{5}\).
05

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:\[r = \frac{\frac{7}{2} \cdot \frac{2}{5}}{1 + \frac{7}{2}\sin(\theta)} = \frac{\frac{7 \times 2}{5 \times 2}}{1 + \frac{7}{2}\sin(\theta)} = \frac{\frac{14}{5}}{1 + \frac{7}{2}\sin(\theta)}.\]
06

Simplify the Polar Equation

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A satellite dish is shaped like a paraboloid of revolution. Th s means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is 12 feet across at its opening and 4 feet deep at its center, where should the receiver be placed?

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=1 ; e=1\)

The mirror in an automobile headlight has a parabolic cross-section with the light bulb at the focus. On a schematic, the equation of the parabola is given as \(x^{2}=4 y .\) At what coordinates should you place the light bulb?

For the following exercises, convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{8}{3-2 \cos \theta} $$

Recall from Rotation of Axes that equations of conics with an \(x y\) term have rotated graphs. For the following exercises, express each equation in polar form with \(r\) as a function of \(\theta\). $$ x^{2}+x y+y^{2}=4 $$
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