Problem 6 - Precalculus: Mathematics for Calculus

Question

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(\frac{1}{2},\) directrix \(y=-4\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{4}{2 + \sin\theta} \).

Step 1: Understand the General Form of a Polar Equation of a Conic

A conic section in polar coordinates, with its focus at the origin, takes the form \( r = \frac{ed}{1 + e\sin\theta} \) or \( r = \frac{ed}{1 + e\cos\theta} \), where \( e \) is the eccentricity and \( d \) is the directrix. The choice between \( \sin \theta \) and \( \cos \theta \) depends on the orientation of the directrix.

Step 2: Analyze the Given Directrix

The given directrix is \( y = -4 \), which implies it is horizontal. Since the directrix is vertical or horizontal, the form of the equation is likely \( r = \frac{ed}{1 + e\sin\theta} \), which is used when the directrix is horizontal.

Step 3: Substitute the Given Eccentricity

For an ellipse, we know \( e < 1 \). The given eccentricity is \( e = \frac{1}{2} \). Substitute this value into the general form, which becomes \( r = \frac{\frac{1}{2}d}{1 + \frac{1}{2}\sin\theta} \).

Step 4: Determine the Directrix Distance

The distance to the directrix \( d \) can generally be derived directly from the specified directrix equation \( y = -4 \). Since it's a horizontal line, the polar form calculation gives \( d = 4 \), due to the magnitude irrespective of its alignment with the x-axis.

Step 5: Substitute Directrix Distance into the Equation

Using \( d = 4 \), substitute back into the equation: \( r = \frac{\frac{1}{2} \times 4}{1 + \frac{1}{2}\sin\theta} = \frac{2}{1 + \frac{1}{2}\sin\theta} \). Multiply numerator and denominator by 2 to simplify, if necessary: \( r = \frac{4}{2 + \sin\theta} \).

Step 6: Final Polar Equation

The polar equation of the ellipse with a focus at the origin, eccentricity \( \frac{1}{2} \), and directrix \( y = -4 \) is \( r = \frac{4}{2 + \sin\theta} \).

Key Concepts

Conic Sections

Conic sections are fascinating figures formed by intersecting a plane with a double-napped cone. These intersections can create different shapes depending on the angle and location of the slice. Conic sections include ellipses, parabolas, hyperbolas, and circles.

• **Ellipse**: An oval shape created when the plane cuts the cone at an angle, but not so steep that it crosses both napes.
• **Parabola**: Formed when the plane cuts parallel to the slope of the cone.
• **Hyperbola**: Occurs when the plane cuts through both napes of the cone, forming two separate curves.
• **Circle**: A special case of an ellipse where the plane cuts the cone perpendicular to its axis.

Understanding these shapes is crucial in fields like physics, engineering, and architecture. Conic sections can be described using different coordinate systems, such as Cartesian or polar coordinates. Polar coordinates are often more convenient for equations centered at the origin, as they express these relationships in terms of distances and angles.

Eccentricity

Eccentricity is a fundamental property of conic sections, indicating how "stretched" or "flattened" a conic is. It's a non-negative real number that defines the shape of the conic section precisely.

• For a **circle**, the eccentricity is 0, as all points are equidistant from the center.
• An **ellipse** has an eccentricity between 0 and 1. It measures how much the ellipse deviates from being a perfect circle.
• A **parabola** has an eccentricity of exactly 1, depicting its distinct U-shaped curve.
• For a **hyperbola**, the eccentricity is greater than 1, reflecting its open, two-piece structure.

The term "eccentricity" helps to categorize the conic sections and influences their equations. For example, in a polar equation of a conic, such as an ellipse, the eccentricity helps relate the distance from a point on the ellipse to its foci and directrix.

Directrix

The directrix is a guiding line used in defining conic sections, intimately linked to their geometric properties. It's significant in framing the relationship between the conic section's focus and any point on the curve.In polar coordinates, the position of the directrix (whether horizontal or vertical) influences the specific form of the conic section’s equation:

• For a **horizontal directrix**, the polar equation usually involves the sine function, as seen in the ellipse equation where the directrix is described as **\(y = -4\)**.
• A **vertical directrix** generally incorporates the cosine function.

By determining the orientation of the directrix, you choose the appropriate trigonometric term (\( heta\) or \( heta - ext{offset}\) ), which then frames how the focus position, eccentricity, and directrix integrate within the polar equation. The directrix provides a foundational line that, alongside eccentricity, meticulously defines the nature of the conic section, ensuring the precision of these mathematical curves when expressed in polar forms.