Problem 33 - Precalculus with Calculus Previews

Question

Identify the given rotated conic. Find the polar coordinates of its vertex or vertices. $$ r=\frac{10}{2-\sin (\theta+\pi / 6)} $$

Step-by-Step Solution

Answer

The conic is a parabola with the vertex at polar coordinates \((1, -\pi/6)\).

Step 1: Identifying the Conic

The given equation \( r = \frac{10}{2 - \sin(\theta + \pi/6)} \) is in the form \( r = \frac{ed}{1 + e \sin(\theta - \theta_0)} \) for a conic section with eccentricity \( e \). We identify the form as an equation of a conic with its focus at the pole, as the angle \((\theta + \pi/6)\) indicates rotation.

Step 2: Determine Eccentricity

The denominator of the conic \( 2 - \sin(\theta + \pi/6) \) reveals it is of the form \( ed - e \sin(\theta + \theta_0) \). Comparing the general conic form and given equation, we find \(e = 1\) since \(e\) is the coefficient of \( \sin \) term, which is 1 here, indicating a parabola (\( e = 1 \)).

Step 3: Position of the Directrix

The constant term \( ed = 2\), indicates that the directrix distance from the pole is \( d = \frac{2}{e} = 2 \) since \( e = 1 \).

Step 4: Find the Vertex

For a parabola with focus at the pole and directrix distance \( d \), the vertex lies at \( r = \frac{d}{2} = 1 \) on the axis of symmetry of the parabola, which confirms the parabola is rotated by \(\pi/6\).

Step 5: Polar Coordinates of the Vertex

Given the rotation \(\theta = \theta_0 - \pi/6 = 0 - \pi/6 = -\pi/6\) at the vertex, the polar angle is adjusted by this rotation, thus the polar coordinates of the vertex are \((1, -\pi/6)\).

Key Concepts

Polar Coordinates

Polar coordinates offer a different way to describe the location of points in a plane compared to the usual Cartesian coordinates. Instead of identifying a point by its x and y values, polar coordinates use a distance from a reference point (the pole, usually akin to the origin in Cartesian terms) and an angle from a reference direction (usually the positive x-axis).

- **Radius (r)**: This component measures how far away the point is from the origin. A larger radius indicates a point farther from the center. - **Angle (θ)**: This part represents the direction corresponding to the distance. It can be measured in radians or degrees and rotates counterclockwise from the positive x-axis.

In the context of conics, like hyperbolas, parabolas, and ellipses, polar coordinates are especially useful for equations that revolve around a focal point, such as in the exercise above. The conic's properties, like its vertices, can succinctly be expressed through these coordinates, making the understanding of rotations and symmetries straightforward.

Eccentricity

Eccentricity is a crucial parameter in understanding conic sections. It defines the shape of a conic section by quantifying how much it deviates from being circular. In mathematical terms, eccentricity (denoted as \(e\)) helps in categorizing conics into:

• **Circles**: Perfect circles have an eccentricity of \(e = 0\). The lowest degree of deviation, or zero deviation, from a circle.
• **Ellipses**: These shapes have \(0 < e < 1\). The more elliptical, the greater the eccentricity, but still less than 1.
• **Parabolas**: Characterized by an eccentricity of \(e = 1\). This specific value indicates that the conic section opens infinitely in one algebraic direction.
• **Hyperbolas**: With \(e > 1\), indicating distinct branches on either side of the cone from which it's derived.

In the specific problem, the eccentricity \(e = 1\) directly identifies the conic as a parabola. Understanding eccentricity enables quick classification of conics, providing insight into their geometric and algebraic properties, as shown in the exercise solution.

Parabola

A parabola is a unique type of conic section which is symmetric about its axis. It can be visualized as the set of all points that maintain equal distance from a fixed point (focus) and a line (directrix). Each parabola has a distinct shape determined by its orientation and stretch, but all parabolas have an eccentricity of exactly 1.

Properties of a parabola include:

• **Focus**: The dot where all points echo an equal distance to the curve itself and the directrix.
• **Directrix**: A line opposite to the direction of focus and symmetrically located.
• **Vertex**: The midpoint of the perpendicular distance between the focus and directrix, representing the parabola's tip or top point, as calculated in polar coordinates in the exercise.

In the exercise given, the parabola is rotated, causing its axis of symmetry to align with the new orientation given by \( \theta + \frac{\pi}{6} \). This rotation affects how we calculate the position of the vertex, showing the importance of understanding how rotations and transformations impact the geometry of conics.