Problem 39 - Algebra and Trigonometry

Question

A polar equation of a conic is given. (a) Find the eccentricity and the directrix of the conic. (b) If this conic is rotated about the origin through the given angle , write the resulting equation. (c) Draw graphs of the original conic and the rotated conic on the same screen. $$ r=\frac{2}{1+\sin \theta} ; \quad \theta=-\frac{\pi}{4} $$

Step-by-Step Solution

Answer

(a) Eccentricity is 1 and directrix is at \( y = -2 \). (b) The rotated equation is \( r = \frac{2}{1 + \frac{\sqrt{2}}{2} (\sin \theta + \cos \theta)} \).

Step 1: Identify the Eccentricity

To find the eccentricity of the conic, compare the given equation \( r = \frac{2}{1 + \sin \theta} \) with the standard form of a polar conic, \( r = \frac{ed}{1 + e\sin \theta} \). Here, the eccentricity \( e \) is 1, implying that the conic is a parabola, since \( e = 1 \) is the characteristic of a parabola.

Step 2: Determine the Directrix

For a polar equation of the form \( r = \frac{ed}{1 + e\sin \theta} \), the length of the directrix \( d \) can be calculated from the numerator. Here, the numerator 2 gives us \( d = 2 \) since the eccentricity \( e = 1 \). Thus, the directrix is a line parallel to the \( x \)-axis located at \( y = -2 \).

Step 3: Rotation of the Conic

To rotate the conic by \( -\frac{\pi}{4} \) about the origin, apply the angle rotation to the polar equation. The new equation after substituting \( \theta \) with \( \theta + \frac{\pi}{4} \) becomes: \[ r = \frac{2}{1 + \sin(\theta + \frac{\pi}{4})} \].

Step 4: Simplify the Rotation Equation

Using the angle sum identity, simplify the new equation: \( \sin(\theta + \frac{\pi}{4}) = \sin \theta \cos \frac{\pi}{4} + \cos \theta \sin \frac{\pi}{4} \). Given \( \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \), the equation becomes: \[ r = \frac{2}{1 + \frac{\sqrt{2}}{2} (\sin \theta + \cos \theta)} \].

Step 5: Graph the Conics

Using plotting software or graphing calculator, plot the original conic: \( r = \frac{2}{1 + \sin \theta} \), which is a parabola, along with the rotated conic \( r = \frac{2}{1 + \frac{\sqrt{2}}{2} (\sin \theta + \cos \theta)} \). The original parabola opens upwards, and the rotated conic is oriented in a direction rotated at \( -\frac{\pi}{4} \).

Key Concepts

Eccentricity

Eccentricity is a fundamental property of conic sections and defines how "stretched" a conic is. In polar coordinates, the eccentricity \( e \) can determine if the conic is a circle, ellipse, parabola, or hyperbola. Here’s a quick guide:

• If \( e = 0 \), the conic is a circle.
• If \( 0 < e < 1 \), it's an ellipse.
• If \( e = 1 \), you have a parabola.
• If \( e > 1 \), it's a hyperbola.

For the given exercise, the equation \( r = \frac{2}{1 + \sin \theta} \) has an eccentricity \( e \) of 1, which classifies it as a parabola. Eccentricity is not just a number; it tells us the shape's openness or curviness in relation to a direct base value in polar form. Understanding eccentricity is key to deciphering the geometric nature and behavior of conic sections.

Directrix

The directrix is a key component of conic sections and works with the focus to define the shape entirely. For a parabola, which is defined as the set of all points equidistant from a point (focus) and a line (directrix), the directrix acts as a reference line. In the polar form \( r = \frac{ed}{1 + e\sin \theta} \), the directrix deviates from the standard horizontal or vertical line seen in Cartesian coordinates.In our exercise, with the numerator of the polar equation being 2 and \( e = 1 \), the directrix is determined to be a line parallel to the x-axis, located at \( y = -2 \). The positioning of the directrix, along with the eccentricity, helps define the exact nature of the parabola encountered.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone. They include ellipses, parabolas, hyperbolas, and circles, each distinguished by their eccentricity value. In analytic geometry, these can be described with equations in Cartesian or polar coordinates.In the context of the exercise, our focused conic is a parabola as the given polar equation demonstrates eccentricity \( e = 1 \). This reflects the unique properties of parabolas among conic sections, including their reflective and geometric characteristics. Conic sections are crucial in various fields, from physics (like planetary motion described by ellipses) to engineering and architecture.

Parabola

A parabola is a unique and open curve among conic sections known for its reflective properties. It's defined as the set of points equidistant from a point (focus) and a line (directrix). With eccentricity equal to 1, parabolas can be modeled both in Cartesian equations and polar coordinates.In our problem, the polar equation \( r = \frac{2}{1 + \sin \theta} \) produced an upward-opening parabola, with its vertex aligned according to the polar axis, highlighting the direction of the symmetric curve. Parabolas are widely used in design and natural phenomena, such as satellite dishes that focus signals to a single point, or the trajectory paths in physics.

Rotation of Axes

Rotation of axes is a process that can simplify the expression of conic sections by aligning them more conveniently relative to the coordinate system. This involves shifting the orientation of the conic in the coordinate plane without distorting its shape.In our exercise, the conic is rotated by \( -\frac{\pi}{4} \) (or -45 degrees) about the origin. This means substituting \( \theta \) with \( \theta + \frac{\pi}{4} \) in the polar equation. Through trigonometric identities, this leads to a modified equation, which helps visualize how the conic appears after the rotation. It's a powerful method for countering angular displacement or simplifying geometry problems.