Question

For the following exercises, find a polar equation of the conic with focus at the origin and eccentricity and directrix as given. $$ \text { Directrix: } \mathrm{y}=2 ; e=2 $$

Short answer

The polar equation is \( r = \frac{4}{1 - 2 \sin \theta} \).

Step-by-step solution

Identify the Conic Type

Since the eccentricity, denoted by \(e\), is 2, we know the conic is a hyperbola because for conics, a hyperbola corresponds to \(e > 1\).

Determine the Form of Equation

For a conic with a directrix parallel to the axes and the focus at the origin, the polar equation can be written as \( r = \frac{ed}{1 - e \sin \theta} \) if the directrix is horizontal (like \(y = 2\)).

Substitute Values

Substitute \(e = 2\) and \(d = 2\) (since the directrix is \(y = 2\)) into the formula from Step 2: \[ r = \frac{2 \times 2}{1 - 2 \sin \theta} = \frac{4}{1 - 2 \sin \theta} \]

Verify the Polar Equation

To make sure, substitute some values of \(\theta\) into the equation \( r = \frac{4}{1 - 2 \sin \theta} \) to confirm the behavior corresponds to a hyperbola by checking that as \(\theta\) approaches values such that the denominator approaches zero, \(r\) tends to infinity. This verifies a hyperbolic shape.

Key concepts

Polar Equations

Polar equations are a way to represent curves using the polar coordinate system. This system is different from the regular Cartesian coordinate system you might be more familiar with. Instead of using x and y to describe a point on a plane, polar coordinates use two values: the radius \( r \) and the angle \( \theta \). The radius is the distance from the origin to the point, while the angle is measured from the positive x-axis.

In the context of conics, polar equations can represent different types of shapes like ellipses, parabolas, and hyperbolas, depending on the eccentricity. The standard form of a polar equation for a conic section with a focus at the origin is:

• If the directrix is horizontal: \( r = \frac{ed}{1 \, \pm \, e \sin \theta} \)
• If the directrix is vertical: \( r = \frac{ed}{1 \, \pm \, e \cos \theta} \)

In these equations, \( e \) is the eccentricity, \( d \) is the distance of the directrix from the origin, and \( \theta \) is the angle in polar coordinates. Using these formulas, you can rewrite given Cartesian equations in polar form, gaining insight into the nature of the conic section.

Conic Sections

Conic sections are curves that are formed by intersecting a plane with a double-napped cone. These include circles, ellipses, parabolas, and hyperbolas. Each of these shapes can be defined in polar coordinates using specific equations. Understanding the characteristics of each can help identify the type of conic:

• Circle: A conic with eccentricity \( e = 0 \).
• Ellipse: Has an eccentricity \( 0 < e < 1 \).
• Parabola: Corresponds to \( e = 1 \).
• Hyperbola: Has an eccentricity \( e > 1 \).

In the exercise about finding a polar equation for a conic, we deal with a hyperbola, since the given eccentricity is 2, which is greater than 1. The identified conic section aligns with the hyperbolic form, determined by the values of the directrix and the eccentricity.

Eccentricity

Eccentricity, often denoted by \( e \), is a fundamental property of conic sections. It describes how "stretched" or "squashed" a conic section appears. The eccentricity value determines the shape of the conic section:

• For a circle: \( e = 0 \).
• For an ellipse: \( 0 < e < 1 \).
• For a parabola: \( e = 1 \).
• For a hyperbola: \( e > 1 \).

Higher eccentricity means more deviation from being circular. In our example, an eccentricity of \( e = 2 \) indicates a hyperbola, confirming its nature as an open curve that expands infinitely. This property is tied closely to the polar equation of the conic, allowing for the transformation of geometric descriptions into analytical expressions based on given eccentricity values.