Problem 30 - 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. $$e=1, \quad y=2$$

Step-by-Step Solution

Answer

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

Step 1: Identify Type of Conic Section

The eccentricity given is \( e = 1 \). In conic sections, when \( e = 1 \), the conic section is a parabola.

Step 2: Write General Formula for Parabola

For conics with one focus at the origin, the polar equation in terms of the eccentricity \( e \) and the directrix is \[ r = \frac{ed}{1 - e\sin\theta} \] if the directrix is vertical, or \[ r = \frac{ed}{1 - e\cos\theta} \] if the directrix is horizontal.

Step 3: Determine Directrix Orientation

Given the directrix is \( y = 2 \), this represents a horizontal line 2 units above the origin.

Step 4: Apply Directrix Orientation to Equation

Since the directrix \( y = 2 \) is horizontal, we use the polar equation \( r = \frac{ed}{1 - e\sin\theta} \).

Step 5: Substitute Values into Equation

Substitute \( e = 1 \) and \( d = 2 \) (the value of \( y \) in the directrix) into the formula: \[ r = \frac{1 \times 2}{1 - 1\sin\theta} \]. This simplifies to \[ r = \frac{2}{1 - \sin\theta} \].

Key Concepts

Eccentricity

When studying conic sections, understanding the concept of eccentricity is crucial. Eccentricity is a number that describes the shape of a conic section. It is usually denoted by the letter \( e \). The value of \( e \) determines whether the conic section is a circle, ellipse, parabola, or hyperbola.

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

In this exercise, the given eccentricity is \( e = 1 \), which indicates that the conic section is a parabola. In polar coordinates, when \( e = 1 \), it implies that the shape bends in such a way that the distance between the points and the directrix is the same as the distance between those points and the focus.

Directrix

The directrix is an important part of defining conic sections. It is a fixed line used in the description of the conic. Each conic has one or more directrices. The role of the directrix is to help compare the distances from a point on the conic to both the focus and the directrix. This comparison defines the shape of the conic section.

In this exercise, the directrix is given as \( y = 2 \). This line is a horizontal line located 2 units above the origin. The equation for a conic section in polar form takes into account the position of the directrix.

• If the directrix is horizontal, the variable \,\( \theta \)\, involves \( \sin \theta \).
• If the directrix is vertical, \( \theta \) involves \( \cos \theta \).

Since the directrix here is horizontal, the polar equation incorporates \( \sin \theta \) in its structure.

Parabola

The parabola is a unique curve that can be represented using polar coordinates when the eccentricity is equal to one. Because of its unique property, a parabola is often used in structures like satellite dishes and airplane wings where the reflective properties are beneficial.

In the context of polar equations, the parabola is characterized by the polar equation:\[ r = \frac{ed}{1 - e\sin\theta}\] where \( e \) is the eccentricity and \( d \) is the perpendicular distance from the focus to the directrix.

• For a horizontal directrix, the equation uses \( \sin \theta \).
• For a vertical directrix, the equation uses \( \cos \theta \).

In our exercise, with \( e = 1 \) and \( d = 2 \), we substitute these into the formula for a parabola with a horizontal directrix, simplifying to the equation:\[ r = \frac{2}{1 - \sin \theta}\] This equation represents the stroke of the parabola in polar coordinates and shows how every point \( r, \theta \) aligns with the geometry of the parabola.