Problem 6 - Precalculus: Mathematical for Calculus

Question

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(0.6,\) directrix \(r=2 \csc \theta\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{1.2}{1 - 0.6\sin\theta} \).

Step 1: Understand the Relationship in Polar Equation

The polar equation for a conic section with its focus at the origin is given by \( r = \frac{ed}{1 - e\sin\theta} \), where \(e\) is the eccentricity and \(d\) is the directrix. Here, the equation of the directrix is \(r = 2 \csc \theta\) which translates into \(d = 2\).

Step 2: Substitute Eccentricity and Directrix

Given \(e = 0.6\) and \(d = 2\), substitute into the formula to get \( r = \frac{0.6 \cdot 2}{1 - 0.6\sin\theta} \). This simplifies to \( r = \frac{1.2}{1 - 0.6\sin\theta} \).

Step 3: Final Polar Equation

The polar equation of the conic that is an ellipse given the conditions is \( r = \frac{1.2}{1 - 0.6\sin\theta} \).

Key Concepts

Conic Sections

Conic sections are curves obtained by slicing a double-napped cone at different angles. They are an integral part of geometry, showing up in various fields like astronomy, engineering, and physics. The four primary types of conic sections are:

• Circle
• Ellipse
• Parabola
• Hyperbola

To better understand, imagine a cone extending infinitely in both directions. The manner in which you cut through the cone determines the type of conic section you get:

• A circle is formed when the cutting plane is parallel to the base of the cone.
• An ellipse results when the plane cuts through the cone at an angle, but not steep enough to pass through both nappes.
• A parabola occurs when the plane is parallel to the slope of the cone.
• A hyperbola arises when the plane intersects both nappes of the cone.

The specific shape of a conic section depends heavily on its eccentricity and the angle of the plane relative to the cone.

Eccentricity

Eccentricity is a crucial parameter that defines the shape of a conic section. It is represented by the symbol \(e\) and determines how "stretched" or "round" the conic is.

• For a circle, \(e = 0\), indicating perfect roundness.
• In an ellipse, \(0 < e < 1\), suggesting it is oval-shaped.
• A parabola has an eccentricity of \(e = 1\).
• A hyperbola has \(e > 1\).

The eccentricity not only classifies the type of conic section but also helps in understanding the behavior of objects in orbital mechanics. In the context of polar coordinates, as used in our exercise, eccentricity helps to calculate the polar equation of the conic. For an ellipse, given \(e = 0.6\), it indicates a moderately elongated shape.

Directrix

The directrix of a conic section is a fixed line used in the formal definition of the curve. The relationship between the directrix, the focus, and any point on the conic leads to the expression of the polar equation.For a conic section, each point \((r, \theta)\) on the curve maintains a constant ratio, \(e\), of its distance from the focus (origin) to its distance from the directrix.In this exercise, the directrix is specified as \(r = 2 \csc \theta\), which simplifies to a linear function. The constant \(d\), which is the distance to the directrix in the polar equation formula \( r = \frac{ed}{1 - e\sin\theta} \), influences how the curve opens and its orientation relative to the coordinate axes.