Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=\frac{4}{1+\cos \theta} $$

Step-by-Step Solution

Answer

The curve is a parabola with eccentricity 1.

Step 1: Identify the form of the polar equation

The given polar equation is \( r = \frac{4}{1+\cos \theta} \). This follows the general form \( r = \frac{ed}{1 + e\cos \theta} \), where \( e \) is the eccentricity, and \( d \) is a constant. In this case, \( ed = 4 \) and the equation is in the standard form of a conic.

Step 2: Determine the eccentricity

From the polar equation \( r = \frac{4}{1+\cos \theta} \), we note that the denominator is \( 1 + e\cos \theta \). Comparing this with \( 1+\cos \theta \), we find that \( e = 1 \). If \( e = 1 \), the conic is a parabola according to conic section properties.

Step 3: Verify the type of conic

A conic section with eccentricity \( e = 1 \) is always a parabola. This confirms that the curve described by the equation \( r = \frac{4}{1+\cos \theta} \) is indeed a parabola.

Step 4: Sketch the graph of the conic

To sketch the graph of the equation \( r = \frac{4}{1+\cos \theta} \), note that it is symmetric with respect to the polar axis because of the \( \cos \theta \) term. The parabola will have its focus at the origin and directrix perpendicular to the axis of symmetry. Since it's opening towards the left (\( -x \) direction), plot points and draw the curve accordingly.

Key Concepts

Eccentricity

Eccentricity is a key concept when analyzing conic sections. For any conic that is expressed in polar coordinates, eccentricity helps determine the shape of the conic. It is denoted by the letter \( e \). The value of \( e \) dictates the nature of the conic section:

• If \( e = 0 \), the conic is a circle.
• If \( 0 < e < 1 \), it is an ellipse.
• If \( e = 1 \), the conic is a parabola.
• If \( e > 1 \), the conic is a hyperbola.

When dealing with the polar equation \( r = \frac{ed}{1 + e\cos\theta} \), the term \( e \) is extracted by comparing the formula with the standard form of given equations. In our exercise, the equation \( r = \frac{4}{1+\cos\theta} \) gave us \( e = 1 \), indicating that this equation describes a parabola.

Conic Sections

Conic sections are curves obtained from the intersection of a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. These shapes can be described using polar equations, which are particularly useful in defining how they stretch and rotate in the coordinate system. In the context of polar coordinates, these curves take on a form \( r = \frac{ed}{1 + e\cos\theta} \) or \( r = \frac{ed}{1 + e\sin\theta} \). This formula not only helps in identifying the conic type but also in determining its orientation and dimension. By identifying the conic from a polar equation, important features like

• symmetric properties,
• focus points,
• and directrix location

can be understood, leading to insight into the structure of the curve.

Parabolas

Parabolas are distinctive shapes in conic sections. They have a unique property: each point on a parabola is equidistant from a fixed point, called the focus, and a fixed line, called the directrix.In our specific exercise, the equation \( r = \frac{4}{1 + \cos \theta} \) forms a parabola. This polar equation signifies that the directrix is perpendicular to the major axis, which in this situation, aligns with the x-axis. The major axis of a parabola is the line that passes through the focus and is perpendicular to the directrix.Parabolas

• are widely used in physics, optics, and engineering,
• often appear when dealing with projectile motion,
• and have reflective properties that are useful in designing telescopes and satellite dishes.

Understanding the geometric and reflective properties of parabolas is crucial in understanding their uses and applications in various fields.