Problem 3 - Algebra and Trigonometry Real Mathematics, Real People

Question

A conic has a polar equation of the form \(r=\frac{e p}{1+e \cos \theta} .\) Is the directrix vertical or horizontal?

Step-by-Step Solution

Answer

The directrix of the conic represented by the given polar equation is horizontal.

Step 1: Understanding the formula

The given polar equation for the conic is \(r=\frac{e p}{1+e \cos \theta}\), where \(r\) is the distance from the origin (focus), \(e\) is the eccentricity, \(p\) is the semi latus rectum (distance from focus to directrix) and \( \theta \) is the angle from the origin. This form of a conic's equation is also known as the polar form of the conic.

Step 2: Analyzing the directrix

The directrix of a conic section is a line perpendicular to the axis of symmetry of the conic (an imaginary line that divides the conic into mirror images). The 'axis of symmetry' is the line from the origin (focus) that makes an angle of \( \theta \) with respect to the x-axis. If \( \theta \) varies between 0 and \( \pi \), then the conic opens upwards or downwards, making the directrix a horizontal line. If \( \theta \) varies between \(- \frac{\pi}{2} \) and \( \frac{\pi}{2} \), the conic opens to the right or left, making the directrix vertical.

Step 3: Determine the direction of the directrix

In the given conic equation \(r=\frac{e p}{1+e \cos \theta}\), since \( \theta \) is not restricted to left and right movement, it means the conic can open upwards or downwards. In this case, the directrix of the given conic is therefore a horizontal line.

Key Concepts

Polar Equations

When dealing with conic sections, expressing them in polar coordinates can simplify the analysis of their shapes and properties. A polar equation describes a curve with respect to a fixed point, known as the pole (similar to the origin in Cartesian coordinates), and a fixed direction. For conic sections, these polar equations often take the form \( r = \frac{ep}{1 + e \cos \theta} \), where \( r \) is the radial distance from the pole, \( \theta \) is the angle with respect to a fixed direction (like the positive x-axis).

This format is beneficial because it directly incorporates important parameters:

• **Eccentricity (\(e\))**: Determines the type of conic section.
• **Semi-latus rectum (\(p\))**: Relates to the physical size of the conic section.

Polar equations adjust naturally to how the conic rotates or changes orientation, which provides a much deeper insight, especially when working with geometric transformations.

Eccentricity

Eccentricity is a crucial parameter in conic sections, influencing both their shape and type. Its value offers a quick way to identify the nature of the conic section, being fundamental in defining the curve:

• **Circle:** If \( e = 0 \), the conic is a perfect circle.
• **Ellipse:** When \( 0 < e < 1 \), the shape is an ellipse—the closer \( e \) is to 0, the more the ellipse resembles a circle.
• **Parabola:** At \( e = 1 \), the conic is a parabola.
• **Hyperbola:** For \( e > 1 \), the conic becomes a hyperbola.

Eccentricity offers a consistent method for distinguishing between the types of conic sections. It determines the "flatness" or "openness" of the curve, providing an intuitive measure of how stretched the conic is from a perfect circle. Conics with different eccentricities also exhibit significant differences in their reflective properties, useful in fields ranging from astronomy to engineering.

Directrix Direction

In conic sections, understanding the directrix's orientation is fundamental for grasping their geometric properties. The directrix is a fixed line, and each point on the conic section maintains a constant ratio of distance to the focus and directrix, governed by the eccentricity \( e \).

The direction of the directrix in the polar equation \( r = \frac{ep}{1 + e \cos \theta} \) is influenced by the variable \( \theta \). The directrix's orientation relates to the conic's axis of symmetry:

• If \( \theta \) allows the conic to open upwards or downwards (often given by angles between 0 and \( \pi \)), the directrix is horizontal.
• If \( \theta \) enables the conic to open to the left or right (typically angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\)), the directrix is vertical.

This distinction is vital, as it allows precise predictions of the conic's behavior. Knowing whether a directrix is vertical or horizontal helps in graphing the conic and understanding the underlying symmetry in its geometry.