Problem 11 - Contemporary Precalculus

Question

Identify the conic section whose equation is given\(;\) if it is an ellipse or hyperbola, state its eccentricity. $$r=\frac{2}{6-4 \cos \theta}$$

Step-by-Step Solution

Answer

Answer: The given polar equation represents a vertical line in rectangular form. Since it is neither an ellipse nor a hyperbola, the question of eccentricity is not applicable in this case.

Step 1: Polar to Rectangular Conversion

To convert the given polar equation to a rectangular form, we'll use the following conversion formulas: $$x = r \cos{\theta}$$ $$y = r \sin{\theta}$$ The given polar equation is: $$r=\frac{2}{6-4 \cos \theta}$$ First, multiply both sides with the denominator to get rid of the denominator: $$r(6-4\cos{\theta})=2$$ Now, we need to substitute \(x\) and \(y\) using the conversion formulas: $$\begin{aligned} (x^2+y^2)(6-4\frac{x}{\sqrt{x^2+y^2}})&=2 \\ (x^2+y^2)(6\sqrt{x^2+y^2}-4x)&=2(x^2+y^2) \end{aligned}$$

Step 2: Simplify the Equation

Now, let's simplify the equation by expanding it and then trying to put it into the standard form of conic sections: $$\begin{aligned} (6\sqrt{x^2+y^2}-4x)(x^2+y^2)&=2(x^2+y^2) \\ 6(x^2+y^2)-4x(x^2+y^2)&=2(x^2+y^2) \\ 4(x^2+y^2)-4x(x^2+y^2)&=0 \\ 4-4x&=0 \end{aligned}$$ Finally, we get a simplified equation: $$4-4x=0$$

Step 3: Identify the Conic Section

From the simplified equation, we can see that this equation represents a vertical line at \(x=1\). A vertical line is neither an ellipse nor a hyperbola, so it has no eccentricity. Therefore, the given polar equation represents a vertical line in rectangular form, and the question of eccentricity is not applicable in this case.

Key Concepts

Polar to Rectangular Conversion

Understanding how to convert equations from polar to rectangular form is crucial for analyzing the characteristics of conic sections. Polar equations use angles and distances from a central point (the pole), whereas rectangular equations are based on coordinate planes with x and y axes. To convert a polar equation, such as the one given in our exercise \(r=\frac{2}{6-4 \cos \theta}\), one can use the identities \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).

During the conversion process, substituting \(r\) with \(\sqrt{x^2+y^2}\) and \(\cos{\theta}\) with \(\frac{x}{r}\) leads to an equation in terms of \(x\) and \(y\). It's essential to remember that \(r\) is always positive, and squaring both sides aids in the elimination of square roots. The primary goal is to manipulate the equation into a familiar form that clearly identifies the conic section in a rectangular coordinate system.

Eccentricity

Eccentricity is a fundamental characteristic of conic sections that describes their shape. It is a non-negative number defining the ratio of the distance from a fixed point (focus) to the distance from a fixed line (directrix). For circles, the eccentricity is 0, which signifies that the radius is constant. For ellipses, the eccentricity ranges from 0 to 1, illustrating their stretched circle shape. For parabolas, the eccentricity is exactly 1, reflecting their unique symmetry and infinite arms. Lastly, for hyperbolas, the eccentricity is greater than 1, showcasing their open, saddle-like curves.

Eccentricity is not difficult to understand but is a key concept in distinguishing between the types of conic sections. It is important not only in mathematics but also in fields such as astronomy, where it is used to describe the orbits of planets and comets around the sun.

Conic Section Identification

After converting a polar equation to rectangular form, identifying the type of conic section represented requires further analysis. Standard forms of conic sections can usually indicate the general shape: \(x^2+y^2=r^2\) for circles, \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) for ellipses, \(y^2=4px\) for parabolas, and \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) for hyperbolas.

Looking into our exercise, after simplification, we reached the equation \(4-4x=0\) which upon further simplification translates to \(x=1\). This does not match any of the standard forms of conic sections and represents a vertical line, a type of degenerate conic section. Unlike traditional conic sections, this vertical line lacks an inherent focus and directrix and thus does not have an eccentricity value.