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Chapter 11: Problem 112

Find the equation in Cartesian coordinates of the lemniscate \(r^{2}=a^{2} \cos 2 \theta,\) where \(a\) is a real number.

Short Answer

Expert verified
Answer: \(x^4 + y^4 - (2a^2 -2)x^2 - a^2y^2 + 2x^2y^2 = 0\)

Step by step solution

01

Convert the polar equation to Cartesian coordinates

To convert the polar equation \(r^2 = a^2\cos2\theta\) to Cartesian coordinates, we need to use the conversion formulas \(x = r\cos\theta\) and \(y = r\sin\theta\). We can do this by first noting that \(r^2 = x^2 + y^2\). Now, we need to find \(\cos2\theta\) in terms of \(x\) and \(y\). We know that: \(\cos2\theta = 2\cos^2\theta - 1 = 2\left(\frac{x}{r}\right)^2 - 1\) since \(\cos\theta = \frac{x}{r}\). Now, we can substitute \(r^2 = x^2 + y^2\) and the expression for \(\cos2\theta\) in the given polar equation: \(x^2 + y^2 = a^2\left(2\left(\frac{x}{\sqrt{x^2 + y^2}}\right)^2 - 1\right)\)
02

Simplify the Cartesian equation

Now we can simplify the above equation as follows: \(x^2 + y^2 = a^2\left(2\left(\frac{x^2}{x^2 + y^2}\right) - 1\right)\) Now let's distribute \(a^2\): \(x^2 + y^2 = 2a^2\left(\frac{x^2}{x^2 + y^2}\right) - a^2\) Next, we will clear the fraction by multiplying both sides of the equation by \(x^2 + y^2\): \((x^2 + y^2)^2 = 2a^2x^2 - a^2(x^2 + y^2)\) Now we can expand the left side of the equation: \(x^4 + 2x^2y^2 + y^4 = 2a^2x^2 - a^2x^2 - a^2y^2\) Finally, we can rearrange the equation to have a more standard form: \(x^4 + y^4 - (2a^2 - 2)x^2 - a^2y^2 + 2x^2y^2 = 0\) Thus, the equation in Cartesian coordinates of the lemniscate is given by: \(x^4 + y^4 - (2a^2 -2)x^2 - a^2y^2 + 2x^2y^2 = 0\)

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Key Concepts

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

Polar Coordinates
Polar coordinates form a two-dimensional coordinate system where each point is determined by a distance from a reference point and an angle from a reference direction. In this system:
  • The reference point is known as the pole, commonly corresponding to the origin in Cartesian coordinates.
  • The reference direction is usually the positive x-axis.
  • The polar coordinates of a point are given as \((r, \theta)\), where \(r\) is the radial distance from the pole, and \(\theta\) is the angle in radians from the reference direction.
Understanding polar coordinates is essential for dealing with shapes like circles, spirals, and lemniscates. These coordinates are particularly useful when the relationship between points involves angles and radial distances.
Lemniscate
A lemniscate is a unique type of curve resembling the shape of a figure-eight or an infinity symbol (鈭�). It is a noteworthy example of a polar graph and is represented in polar coordinates by the equation:\[r^2 = a^2 \cos 2\theta\]where \(a\) is a constant. The lemniscate exhibits interesting properties:
  • It is symmetric about the polar axis and the origin.
  • Its shape is derived from the function \(\cos 2\theta\), causing it to cross itself, creating the iconic figure-eight form.
Lemniscates appear in various scientific contexts, including physics, engineering, and computational geometry. Its study involves both polar and Cartesian coordinates for complete analysis.
Equation Conversion
Equation conversion between polar and Cartesian coordinates is a highly useful skill, especially for graphing complex curves like lemniscates. When converting, two primary formulas are often used:
  • \(x = r\cos\theta\)
  • \(y = r\sin\theta\)
Additionally, the relationship \(r^2 = x^2 + y^2\) is crucial for transforming the equations. For instance, converting the lemniscate's equation from polar to Cartesian involves replacing \(r\) and \(\cos2\theta\) with expressions involving \(x\) and \(y\). This necessitates the use of trigonometric identities and algebraic manipulation, allowing the intricate curves defined in polar form to be expressed in the more familiar Cartesian plane.
Trigonometric Identities
Trigonometric identities are pivotal in simplifying and converting equations between coordinate systems. Identities involving double angles, such as \(\cos 2\theta = 2\cos^2\theta - 1\), help transform expressions involving trigonometric functions to a form suitable for Cartesian manipulation. In the context of the lemniscate:
  • Rewriting \(\cos\theta\) as \(\frac{x}{r}\) allows the substitution into the double angle identity.
  • These identities enable the reshaping of the curve's equation, revealing its structure and behavior in Cartesian coordinates.
Mastering these identities is essential for solving problems in calculus, physics, and engineering, where switching between polar and Cartesian perspectives frequently occurs.

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Most popular questions from this chapter

Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. $$r=\frac{1}{2-2 \sin \theta}$$

Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work. A hyperbola with vertices (卤1,0) that passes through \(\left(\frac{5}{3}, 8\right)\)

Without using a graphing utility, determine the symmetries (if any) of the curve \(r=4-\sin (\theta / 2)\)

Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola that opens to the right with directrix \(x=-4\)

Find an equation of the following parabolas, assuming the vertex is at the origin. A parabola with focus at (-4,0)
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