Problem 29 - Thomas Calculus

Question

Exercises \(29-36\) give the eccentricities of conic sections with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$ e=1, \quad x=2 $$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{2}{1 + \cos(\theta)} \).

Step 1: Understand the Elements of the Conic

We have an eccentricity \( e = 1 \) which indicates that the conic section is a parabola. The directrix given is \( x = 2 \). For a parabola, the definition involves its focus, given as the origin \((0,0)\), and this directrix.

Step 2: Recall the Equation Form for a Parabola

The general polar equation for a conic section is \( r = \frac{ed}{1 + e\cos(\theta)} \) or \( r = \frac{ed}{1 + e\sin(\theta)} \) depending on the orientation. For a parabola with a vertical directrix \( x = d \), the equation simplifies to \( r = \frac{d}{1 + e\cos(\theta)} \) given \( e = 1 \).

Step 3: Substitute the Values into the Equation

Substitute the values \( e = 1 \) and \( d = 2 \) into the formula: \[ r = \frac{1 \cdot 2}{1 + 1 \cos(\theta)} = \frac{2}{1 + \cos(\theta)}. \]

Step 4: Simplify the Polar Equation

Since the equation already simplified to \( r = \frac{2}{1 + \cos(\theta)} \), this is the final polar equation for the parabola with a focus at the origin and directrix at \( x = 2 \).

Key Concepts

Eccentricity

Eccentricity is a fundamental concept when studying conic sections, which include parabolas, ellipses, and hyperbolas. It determines the shape and type of the conic section. The eccentricity \( e \,\) of a conic section is a non-negative real number that tells us how much the conic deviates from being circular.

• For a circle, the eccentricity is \( e = 0\).
• An ellipse has an eccentricity such that \( 0 < e < 1\).
• A parabola is defined by an eccentricity of \( e = 1\).
• A hyperbola has an eccentricity \( e > 1\).

In our specific exercise, the eccentricity \( e = 1\) signifies the curve is a parabola. This means that there is a single focus at the origin and the directrix that guides the definition of the parabola's shape. Understanding eccentricity helps predict the type of curve and provides insights into the behavior and properties of different conic sections.

Parabola

A parabola is a specific type of conic section where each point on the curve is equidistant from a fixed point called the focus and a line called the directrix. In our exercise, the focus is at \( (0,0) \,\) and the directrix is the vertical line \( x = 2\).To find the polar equation of a parabola, you can use the formula: \[ r = \frac{ed}{1 + e\cos(\theta)} \] or equivalent formulas depending on orientation. Here, \( e\) stands for eccentricity, and \( d\) represents the perpendicular distance from the origin (the pole) to the directrix.For parabolas, because \( e = 1\), the polar equation simplifies to ewline \[ r = \frac{d}{1 + \cos(\theta)} \] for a vertical directrix. As seen in our exercise, this yields the final equation \( r = \frac{2}{1+\cos(\theta)}\).This equation reflects the nature of parabolas to be open and at an equal distance at any point from the focus and the directrix. Learning about parabolas not only links geometry with algebra but also opens doors to understanding their applications.

Directrix

In the context of conic sections, the directrix is a crucial line that, along with the focus, helps define the shape of the curve. For a parabola, the role of the directrix is particularly important.The directrix in our exercise is given by \( x = 2\), which means it is a vertical line placed on the Cartesian plane. It is situated to the right of the focus at \( (0,0) \,\)The directrix works with the focus to maintain the unique property of the parabola. Every point \( (r, \theta) \,\) on the parabola maintains a consistent ratio of distances measured from the focus and perpendicular to the directrix. This ratio is precisely the eccentricity, which here is \( e = 1\).

• For points on the parabola, the distance to the focus equals the distance to the directrix.
• The directrix helps ensure that the shape remains consistent as a parabola.

Recognizing the importance of the directrix in defining conics helps in solving problems involving parabolic curves and extends to similar considerations for ellipses and hyperbolas.