Problem 51 - Calculus Early Transcendentals

Question

Show that the polar curve \(r=4+2 \sec \theta\) (called a conchoid) has the line \(x=2\) as a vertical asymptote by showing that $$\lim _{r \rightarrow \pm \infty} x=2 .$$ Use this fact to help sketch the conchoid.

Step-by-Step Solution

Answer

The conchoid has a vertical asymptote at \(x=2\).

Step 1: Convert Polar to Cartesian Coordinates

The polar curve is given by \( r = 4 + 2\sec \theta \). To find \( x \) in Cartesian coordinates, use the relation \( x = r\cos\theta \).\ Since \( \sec\theta = \frac{1}{\cos\theta} \), \( r \cos\theta = (4 + 2\sec\theta)\cos\theta = 4\cos\theta + 2 \). Thus, \( x = 4\cos\theta + 2 \).

Step 2: Investigate the Limiting Behavior

We need to find \( \lim_{r \rightarrow \pm \infty} x \).\ From the expression \( x = 4\cos\theta + 2 \), where \( \cos\theta = \frac{1}{\sec\theta} \), as \( \sec \theta \to \pm \infty \), \( \cos \theta \) approaches 0.\ Substitute to get \( x = 4 \times 0 + 2 = 2 \).

Step 3: Sketching the Asymptote

Since we showed that \( \lim_{r \rightarrow \pm \infty} x = 2 \), the line \( x = 2 \) is a vertical asymptote of the conchoid.\ In the sketch of the conchoid, draw the line \( x = 2 \) to indicate where the curve approaches vertically.

Key Concepts

Conchoid

The conchoid is an intriguing type of polar curve, with a unique shape characterized by its mathematical relationship with a fixed point, often creating loops or cusp features. The specific conchoid given by the equation \( r = 4 + 2 \sec \theta \) shifts from traditional polar representations. Here, the term \( 2 \sec \theta \) introduces a fascinating behavior as \( \theta \) varies.

• The term \( \sec \theta \) (the reciprocal of \( \cos \theta \)) indicates how the radius stretches or shrinks drastically around certain angles.
• The constant "4" ensures a baseline distance from the origin, influencing how the curve unfolds.

Understanding this polar curve's behavior, especially when visualized, emphasizes its neat asymptotic properties and the effect of the secant function on its shape.

Vertical asymptote

Vertical asymptotes are lines that a curve approaches but never actually touches or crosses. For the conchoid given by \( x = r \cos \theta = 4 \cos \theta + 2 \), we want to examine when \( x \) approaches a specific value as \( r \rightarrow \pm \infty \).

• By setting up the limit \( \lim_{r \rightarrow \pm \infty} x = 2 \), we investigate the behavior as \( \sec \theta \) diverges.
• As \( \sec \theta \) becomes increasingly large, \( \cos \theta \) tends toward zero, simplifying \( x \) directly to 2.

Therefore, the line \( x = 2 \) acts as a vertical asymptote for this conchoid. It serves as a boundary the curve approaches vertically but never reaches, providing insight into the conchoid's sweeping path.

Polar to Cartesian Conversion

Converting from polar to Cartesian coordinates can help in the visualization and analysis of polar curves. In our exercise, we started with the polar equation \( r = 4 + 2 \sec \theta \) and converted it to a Cartesian equation.

• The conversion process uses the basic relationship: \( x = r \cos \theta \).
• Substituting in our given equation, \( r \cos \theta = 4 \cos \theta + 2 \), gives us a Cartesian form that depends on \( \cos \theta \).
• This equation simplifies the analysis and sketching of the curve by working within the familiar Cartesian plane.

By translating polar equations into Cartesian coordinates, we can interpret the curve's graphical traits, allowing us to discover key geometrical features like asymptotes and other boundary behaviors.