Problem 46 - College Algebra

Question

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(y=-2 ; e=\frac{1}{2}\)

Step-by-Step Solution

Answer

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

Step 1: Understand Polar Conics

In polar coordinates, a conic section with focus at the origin can be expressed as \( r = \frac{ed}{1 + e\sin(\theta)} \) when the directrix is horizontal and \( y = c \). Here, \( r \) is the radius, \( e \) is the eccentricity, and \( d \) is the distance from the focus to the directrix.

Step 2: Identify Given Values

From the problem, we are given the eccentricity \( e = \frac{1}{2} \) and the directrix \( y = -2 \). This means the directrix is horizontal, so we use the equation for a horizontal directrix.

Step 3: Calculate Distance to Directrix

The directrix is \( y = -2 \), which means it is \( 2 \) units below the origin. So, the distance \( d = 2 \). Since the directrix is below the focus, this distance is positive in the equation.

Step 4: Substitute Values into Equation

Substitute the values of \( e = \frac{1}{2} \) and \( d = 2 \) into the polar equation \( r = \frac{ed}{1 + e\sin(\theta)} \) to get:\[r = \frac{\left(\frac{1}{2}\right)2}{1 + \left(\frac{1}{2}\right)\sin(\theta)} = \frac{1}{1 + \frac{1}{2}\sin(\theta)}\]

Step 5: Simplify the Equation

To simplify further, multiply the numerator and the denominator by 2 to eliminate the fraction in the denominator:\[r = \frac{2}{2 + \sin(\theta)}\]This is the polar equation of the conic given the problem's parameters.

Key Concepts

Eccentricity

Eccentricity is a key characteristic that defines the shape of a conic section. It is represented by the symbol \( e \). In simple terms, eccentricity measures how much a conic section deviates from being circular.

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

In our example, the eccentricity \( e = \frac{1}{2} \), which is less than 1, indicating that the conic section is an ellipse. Eccentricity helps us understand how stretched or elongated an ellipse is. The closer \( e \) is to zero, the more the shape resembles a circle. This value is crucial in deriving the conic's equation, as it influences the terms involving trigonometric functions like sine or cosine.

Directrix

The directrix is a critical line in the geometric construction of conic sections. It is used in conjunction with the eccentricity to define the conic's equation in polar coordinates.
The position of the directrix directly affects the shape and orientation of the conic section.

• For a horizontal directrix like \( y = -2 \), it is a line parallel to the x-axis.
• The distance \( d \) from the origin to this line was determined to be 2 units because the directrix lies below the focus at the origin.

The directrix is fundamental in setting up the polar equation. In our problem, knowing the position of the directrix allows us to use the polar conic formula.

Polar Coordinates

Polar coordinates are used to describe the position of points in a plane using a radius and an angle. These coordinates are especially useful for describing conic sections centered at the origin.
The polar coordinate system helps us express conic sections in a simpler form by using the following parameters:

• \( r \) - the radius or distance from the origin.
• \( \theta \) - the angle from the positive x-axis to the line connecting the origin to the point in question.

In our exercise, we expressed the conic using the polar equation \( r = \frac{2}{2 + \sin(\theta)} \). Understanding how polar coordinates work is essential for comprehending how the different elements (eccentricity and directrix) affect the equation and the shape of the conic.

Conic Sections

Conic sections are the curves obtained by slicing a cone with a plane. They include ellipses, parabolas, and hyperbolas. Each type has distinct characteristics determined by its eccentricity.
These shapes can be described in Cartesian coordinates, but polar coordinates often provide a more straightforward formulation.

• Ellipses have a constant eccentricity \( 0 < e < 1 \).
• Parabolas have an eccentricity \( e = 1 \).
• Hyperbolas have an eccentricity \( e > 1 \).

In polar coordinates, equations can succinctly represent these shapes:\[ r = \frac{ed}{1 + e\sin(\theta)} \] for horizontal directrices or \[ r = \frac{ed}{1 + e\cos(\theta)} \] for vertical directrices. Recognizing the conic from its equation allows you to understand its properties and graph it effectively. The given problem reveals that the conic is an ellipse due to its eccentricity of \( \frac{1}{2} \), aligning with its shape properties.