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Chapter 10: Problem 65

Graph the lines and conic sections. $$r=3 \sec (\theta-\pi / 3)$$

Short Answer

Expert verified
The line is vertical at \(x=3\) in Cartesian coordinates.

Step by step solution

01

Convert Polar to Cartesian Form

The given polar equation is \[ r = 3 \sec \left( \theta - \frac{\pi}{3} \right) \] First, recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} \). This implies \[ r = \frac{3}{\cos \left( \theta - \frac{\pi}{3} \right) } \]. This corresponds to a line because secant functions in polar coordinates indicate vertical lines. To convert this to a Cartesian equation, use the identities: \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). We also have the identity \[ x = 3. \] This is the horizontal line at \(x=3\).
02

Plot the Cartesian Equation

Now that we have the Cartesian version of the equation, which is simply \(x=3\), plot this equation on a coordinate grid.This represents a vertical line passing through the x-axis at \(x = 3\). The line extends vertically in both directions across the y-axis. It is important to note that all points on the line have an x-coordinate of 3, signifying that the line does not vary in the x direction.

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Key Concepts

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

Graphing Lines
Graphing lines is a fundamental aspect of understanding coordinate systems. Lines in Cartesian coordinates are represented as linear equations like \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. A vertical line, like the one we discussed with \(x = 3\), is unique because it doesn't vary in the x-direction; its equation features invariable x-values. This means wherever you find yourself on the line, the x-coordinate remains consistent.

When graphing a line such as \(x = 3\), include:
  • Plotting points along the x-coordinate 3
  • Drawing a straight vertical line extending through points at x = 3 on both positive and negative y-directions
Visualizing lines like this facilitates understanding spatial relationships in the Cartesian plane, bridging concepts for more complex graphing, including polar coordinates.
Conic Sections
Conic sections are the curves obtained by slicing a right circular cone with a plane. There are four main types: circles, ellipses, parabolas, and hyperbolas. Each has distinct properties and equations:
  • Circles: All points are at a constant distance from the center. The equation is \((x-h)^2 + (y-k)^2 = r^2\).
  • Ellipses: Similar to circles but stretched along one axis. An ellipse's equation is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\).
  • Parabolas: Curves where any point is the same distance from a fixed point (focus) and a line (directrix). The equation is usually \(y = ax^2 + bx + c\).
  • Hyperbolas: Two similar open curves that are mirror images. The typical equation is \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\).
Understanding conic sections helps in visualizing and interpreting different types of curves. They're significant in fields like engineering and physics, where they appear frequently in natural phenomena and design.
Polar Coordinates
Polar coordinates represent points in a plane using a radius and angle, denoted as \( (r, \theta) \). This system is particularly effective in scenarios involving circular and radial patterns, as it reflects angles and distances from a central point.

To convert between polar and Cartesian coordinates, employ the following relationships:
  • Circular Systems: Defined by the center and angle. Begin with the polar coordinates \(r = 3 \sec(\theta - \frac{\pi}{3})\), which implies a line in polar form.
  • Conversion to Cartesian: Utilize \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) to translate polar equations into Cartesian. Thus, polar lines transition into vertical or horizontal lines in Cartesian coordinates.
Grasping polar coordinates simplifies the representation and analysis of systems with rotational symmetry, aiding in visualizing equations and lines not easily interpreted in Cartesian form.

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Most popular questions from this chapter

Replace the Cartesian equations with equivalent polar equations. $$x^{2}+(y-2)^{2}=4$$

Give the foci or vertices and the eccentricities of ellipses centered at the origin of the \(x y\) -plane. In each case, find the ellipse's standard-form equation in Cartesian coordinates. Vertices: (卤10,0) Eccentricity: 0.24

Find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices. $$y^{2}-3 x^{2}=3$$

Replace the Cartesian equations with equivalent polar equations. $$y=1$$

Give the eccentricities and the vertices or foci of hyperbolas centered at the origin of the \(x y\) -plane. In each case, find the hyperbola's standard-form equation in Cartesian coordinates. Eccentricity: Vertices: (0,卤1)
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