Problem 35 - 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. $$e=1 / 5, \quad y=-10$$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{10}{5 + \sin\theta} \).

Step 1: Understand the Problem Components

The given eccentricity is \(e = \frac{1}{5}\) and the directrix is \(y = -10\). We need to write the polar equation of a conic section with one focus at the origin, based on these values.

Step 2: Identify the Type of Conic Section

The eccentricity \(e < 1\) indicates that the conic is an ellipse. Ellipses have eccentricities between \(0\) and \(1\).

Step 3: Formula for Polar Equation of an Ellipse

The polar equation of a conic with eccentricity \(e\) and directrix \(r = d\) is given by \[ r = \frac{ed}{1 + e\sin\theta} \] or \[ r = \frac{ed}{1 + e\cos\theta} \], depending on the orientation of the directrix. For horizontal directrix \(y = -10\), use \(\sin\theta\).

Step 4: Substitute the Values

For this problem, \(e = \frac{1}{5}\) and \(d = 10\) since the directrix is horizontal at \(y = -10\). Substitute these into the equation: \[ r = \frac{(1/5) \cdot 10}{1 + (1/5)\sin\theta} \]

Step 5: Simplify the Equation

Simplify the equation: \[ r = \frac{2}{1 + \frac{1}{5} \sin\theta} \]. Multiply numerator and denominator by 5 to clear the fraction: \[ r = \frac{10}{5 + \sin\theta} \]

Step 6: Final Polar Equation

Thus, the polar equation of the ellipse is \[ r = \frac{10}{5 + \sin\theta} \]. This represents the ellipse with the given eccentricity and directrix.

Key Concepts

Eccentricity

Eccentricity is a key characteristic that defines the shape of conic sections.It is represented by the symbol \(e\).
In simple terms, eccentricity measures how much a conic section deviates from being a circle.The value of eccentricity helps determine the type of conic:

• If \(e = 0\), the conic section is a circle.
• If \(0 < e < 1\), the conic is an ellipse, as seen in the current exercise.
• If \(e = 1\), the conic is a parabola.
• If \(e > 1\), the conic becomes a hyperbola.

In our problem, the given eccentricity is \(e = \frac{1}{5}\), thus classifying the conic section as an ellipse.This value means the ellipse is closer to a circle compared to a conic with a higher eccentricity value.

Conic Sections

Conic sections are the curves obtained by the intersection of a plane and a double-napped cone. Depending on the angle and position of the intersection, different types of conic sections are formed.
The main types include:

• Circles - formed when the plane cuts perpendicular to the cone's axis.
• Ellipses - result from the plane cutting at an angle less steep than the cone.
• Parabolas - arise when the plane is parallel to the cone's slant height.
• Hyperbolas - occur when the plane cuts through both nappes of the cone.

Each type of conic section has unique equations in both Cartesian and polar forms. In polar coordinates, these equations incorporate eccentricity to describe their curves more accurately. For instance, an ellipse has a polar equation format that reflects its eccentricity and the orientation of its directrix.

Ellipse

An ellipse is one of the fundamental types of conic sections.It can be visualized as an elongated circle.
The defining property of an ellipse is that the sum of distances from any point on the ellipse to two fixed points, known as the foci, is constant.This property leads to its oval shape.
Ellipses have unique characteristics:

• They have two axes: the major axis (longest diameter) and minor axis (shortest diameter).
• The center of the ellipse is the midpoint between the two foci.
• In polar coordinates, an ellipse is described with respect to one focus, typically at the origin.

In our exercise, the eccentricity (\(e < 1\)) confirms that the conic is indeed an ellipse.The given directrix and polar equation form further help determine its exact graphical representation.

Polar Coordinates

Polar coordinates offer a method of representing points in a plane through a radius and angle, rather than Cartesian x and y coordinates.This system is particularly useful for conic sections when forms are aligned with a focus at the origin.
The basic components of polar coordinates include:

• \(r\) - the radial distance from the origin to the point.
• \(\theta\) - the angle formed with the positive x-axis.

Converting conic sections into polar equations allows for simplified expressions that inherently accommodate the location of the focus, which is convenient compared to Cartesian expressions.
For example, in this exercise, the polar equation form effectively communicates the relationship between the directrix and the eccentricity, enabling precise description of the ellipse shape.Polar coordinates are indispensable in fields like physics and engineering due to their ability to succinctly represent angles and distances.