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Chapter 8: Problem 187

Convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented. \(r=2 \sec \theta\)

Short Answer

Expert verified
The Cartesian equation is \( x = 2 \), representing a vertical line.

Step by step solution

01

Recall the Polar to Cartesian Conversion Formulas

To convert from polar to Cartesian coordinates, remember that the formulas are: \ \( x = r \cos \theta \) and \( y = r \sin \theta \). Also, \( r^2 = x^2 + y^2 \).
02

Express Secant in Terms of Cosine

Recall that \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, the given equation \( r = 2 \sec \theta \) can be rewritten as \( r = \frac{2}{\cos \theta} \).
03

Multiply to Clear the Fraction

Multiply both sides by \( \cos \theta \) to clear the fraction: \ \( r \cos \theta = 2 \).
04

Substitute for Cartesian Coordinates

Replace \( r \cos \theta \) with \( x \) using the polar to Cartesian conversion, thus the equation becomes: \ \( x = 2 \).
05

Identify the Conic Section

The equation \( x = 2 \) is a vertical line. It does not represent a conic section, as conic sections are typically ellipses, parabolas, hyperbolas, and circles. However, a line can be considered a degenerate conic.

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Key Concepts

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

Conic Sections
Conic sections are fascinating curves that arise from slicing a cone with a plane. There are four primary types of conic sections: the circle, ellipse, parabola, and hyperbola. Each type has unique characteristics:
  • Circle: All points are equidistant from a single point, called the center.
  • Ellipse: Similar to a circle, but with two focal points. All points on the ellipse have a constant sum of distances to the foci.
  • Parabola: This has one focus and a directrix. The distance from any point on the parabola to the focus equals its distance to the directrix.
  • Hyperbola: Comprises two separate curves. The difference in distances from any point on a hyperbola to two foci is constant.
In solving conic section problems, understanding the geometric properties and equations that define each conic type is essential. In this exercise, the equation resulted in a vertical line, which we refer to as a degenerate conic, showing how conics can relate to simpler geometric shapes.
Cartesian Coordinates
Cartesian coordinates serve as a foundation in mathematics, allowing us to map points in space using two intersecting axes. Originating from the work of Ren茅 Descartes, this system uses ordered pairs
  • X-Axis: The horizontal axis, running left to right in a plane.
  • Y-Axis: The vertical axis, running top to bottom in a plane.
  • Coordinate Pair: Represents a point by its horizontal (x) and vertical (y) displacement from the origin (0,0).
For example, the point (3, 4) indicates a position 3 units to the right and 4 units up from the origin. In our exercise, we used the Cartesian coordinate formula to transform the polar equation into a form that can be graphed on a Cartesian plane, specifically landing on the line x = 2. This process clarifies how lines and conics can be represented in a straightforward, visual manner.
Polar Coordinates
Polar coordinates offer a different take on locating points. Instead of relying on horizontal and vertical displacements, they use angles and distances from a fixed point, known as the pole, akin to the origin in Cartesian coordinates.
  • Radial Distance (r): The distance from the pole to the point.
  • Angle (胃): Measured from a reference direction, typically the positive x-axis.
Consider the point given as (r, 胃). Here, 'r' specifies how far the point is from the origin, and '胃' describes the angle from the horizontal axis. The transformation between polar and Cartesian coordinates involves using the formulas:
  • Conversion to x: \( x = r \cos \theta \)
  • Conversion to y: \( y = r \sin \theta \)
In our exercise, these conversions allowed us to find that a "fancy" polar equation actually simplifies to a straight line in the Cartesian system.

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Most popular questions from this chapter

As part of avideo game, the point \((7,3)\) is rotated counterclockwise about the origin through an angle of \(40^{\circ} .\) Find the new coordinates of this point.

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