Problem 112
Question
Find the equation in Cartesian coordinates of the lemniscate \(r^{2}=a^{2} \cos 2 \theta,\) where \(a\) is a real number.
Step-by-Step Solution
Verified Answer
Question: Determine the equation of the lemniscate given its polar equation \(r^{2} = a^{2} \cos 2 \theta\) in Cartesian coordinates.
Answer: The equation of the lemniscate in Cartesian coordinates is \(x^4 - a^2x^2 + x^2y^2 + y^4 - a^2y^2 = 0\).
1Step 1: Write down the given polar equation
We are given the polar equation: \(r^{2}=a^{2} \cos 2 \theta\)
2Step 2: Convert the equation into Cartesian coordinates
We'll use the conversion formulas: \(x = r \cos \theta\) and \(y = r \sin \theta\). The given equation can also be written as: \(r^2 = a^2 \cos 2\theta\).
We also know that \(r^2 = x^2 + y^2\) and \(\cos 2\theta = 1 - 2 \sin^2 \theta\). To get \(\sin^2 \theta\), we can use \(r = \sqrt{x^2 + y^2}\) and \(\sin \theta = \frac{y}{r}\). So, \(\sin^2 \theta = \frac{y^2}{x^2 + y^2}\).
Substituting these expressions into the given equation, we get:
\(x^2 + y^2 = a^2(1 - 2\frac{y^2}{x^2 + y^2})\)
3Step 3: Simplify the equation
Now let's simplify the equation by clearing the denominator and organizing the terms:
\((x^2 + y^2)^2 = a^2(x^2 + y^2) - 2a^2y^2\)
\(x^4 + 2x^2y^2 + y^4 = a^2x^2 + a^2y^2 - 2a^2y^2\)
\(x^4 + 2x^2y^2 + y^4 - a^2x^2 - a^2y^2 + 2a^2y^2 = 0\)
Combining the \(x^2y^2\) and \(y^2\) terms, we get:
\(x^4 - a^2x^2 + x^2y^2 + y^4 - a^2y^2 = 0\)
Thus, the equation of the lemniscate in Cartesian coordinates is:
\(x^4 - a^2x^2 + x^2y^2 + y^4 - a^2y^2 = 0\)
Key Concepts
Polar to Cartesian conversionTrigonometric identitiesAnalytic geometry
Polar to Cartesian conversion
Converting equations from polar to Cartesian coordinates is a common step in mathematics, especially in analytic geometry. The polar coordinate system defines a point by the distance from the origin and the angle from a reference direction. In contrast, the Cartesian coordinate system represents a point by its horizontal and vertical distances from the origin using x and y axes.
To convert a polar equation like a lemniscate, given by \(r^{2}=a^{2} \cos 2 \theta\), to Cartesian form, we use the relationships:
\[ \begin{align*} x &= r \cos \theta, \ y &= r \sin \theta \ \end{align*} \]
Additionally, the identity \(r^2 = x^2 + y^2\) is crucial here. Substituting these into the original polar equation allows us to express the equation entirely in terms of x and y, yielding a form suitable for Cartesian analysis.
This conversion doesn't just change the format of an equation; it often reveals symmetries and other properties that aren't apparent in polar form.
To convert a polar equation like a lemniscate, given by \(r^{2}=a^{2} \cos 2 \theta\), to Cartesian form, we use the relationships:
\[ \begin{align*} x &= r \cos \theta, \ y &= r \sin \theta \ \end{align*} \]
Additionally, the identity \(r^2 = x^2 + y^2\) is crucial here. Substituting these into the original polar equation allows us to express the equation entirely in terms of x and y, yielding a form suitable for Cartesian analysis.
This conversion doesn't just change the format of an equation; it often reveals symmetries and other properties that aren't apparent in polar form.
Trigonometric identities
Trigonometric identities are equations that hold for all values of the variables within their domain. They are essential tools for simplifying expressions and solving trigonometric equations.
Here, we used the identity for \(\cos 2\theta\), which is expressed as:
\[ \cos 2\theta = 1 - 2 \sin^2 \theta \]
This identity helps us express the cosine term in terms of sine, which is often necessary when working with Cartesian coordinates. In our solution, substituting \(\sin^2 \theta = \frac{y^2}{x^2 + y^2}\) enabled us to rewrite the original polar equation in a form that relates directly to x and y.
Understanding these identities is crucial when you need to manipulate or combine trigonometric expressions, making problems easier to handle and solutions more straightforward to reach.
Here, we used the identity for \(\cos 2\theta\), which is expressed as:
\[ \cos 2\theta = 1 - 2 \sin^2 \theta \]
This identity helps us express the cosine term in terms of sine, which is often necessary when working with Cartesian coordinates. In our solution, substituting \(\sin^2 \theta = \frac{y^2}{x^2 + y^2}\) enabled us to rewrite the original polar equation in a form that relates directly to x and y.
Understanding these identities is crucial when you need to manipulate or combine trigonometric expressions, making problems easier to handle and solutions more straightforward to reach.
Analytic geometry
Analytic geometry, also known as coordinate geometry, is the study of geometry using a coordinate system where algebra is employed to solve geometric problems. It merges algebraic equations with geometric figures.
The lemniscate, a classic figure in analytic geometry, has the polar equation \(r^{2}=a^{2} \cos 2 \theta\). The task of converting this into Cartesian coordinates deals directly with understanding how geometric curves can be represented algebraically.
In Cartesian coordinates, the resultant equation \(x^4 - a^2x^2 + x^2y^2 + y^4 - a^2y^2 = 0\) specifies the shape and location of the lemniscate on the xy-plane. This representation in Cartesian form is valuable because it allows us to apply various algebraic techniques to analyze and understand the geometric figure's properties, such as symmetry and intercepts.
Analytic geometry facilitates a deeper understanding of the nature of geometric shapes by uniting algebra and geometry, enabling the graphical representation of algebraic equations and vice versa.
The lemniscate, a classic figure in analytic geometry, has the polar equation \(r^{2}=a^{2} \cos 2 \theta\). The task of converting this into Cartesian coordinates deals directly with understanding how geometric curves can be represented algebraically.
In Cartesian coordinates, the resultant equation \(x^4 - a^2x^2 + x^2y^2 + y^4 - a^2y^2 = 0\) specifies the shape and location of the lemniscate on the xy-plane. This representation in Cartesian form is valuable because it allows us to apply various algebraic techniques to analyze and understand the geometric figure's properties, such as symmetry and intercepts.
Analytic geometry facilitates a deeper understanding of the nature of geometric shapes by uniting algebra and geometry, enabling the graphical representation of algebraic equations and vice versa.
Other exercises in this chapter
Problem 110
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