Problem 40 - Precalculus with Limits

Question

In Exercises 39-54, find a polar equation of the conic with its focus at the pole. \(\textit{Conic}\) Parabola \(\textit{Eccentricity}\) \(e=1\) \(\textit{Directrix}\) \(y=-4\)

Step-by-Step Solution

Answer

The polar equation of the conic is \(r = 4(1 - r)\).

Step 1: Determine the distance from the directrix and the pole

The distance from the directrix to the pole is determined using the formula \(d = |c/a|\), where \(c\) is the constant in the equation \(y = mx + c\) and \(a\) is the number in front of \(y\). In this case, the equation of the directrix is \(y = -4\), hence \(d = |(-4)/1| = 4\). This is also the semi-latus rectum \(l\) in the formula for polar equations of conics.

Step 2: Determine the Orientation

The orientation of the conic relative to the pole where the focus of the parabola is, is along the positive y-axis. However, according to the polar equation template \(r=\frac{l}{1+e\cos(\theta)}\) the \(cos\) function indicates a movement along x-axis. Hence, \(\theta\) makes an angle 90 degrees counter-clockwise from the x-axis i.e., \(-ϴ\) is replaced with \(-(ϴ-90°) or (ϴ+90°)\).

Step 3: Substitute the Values

Substitute the value of \(e\) and \(l\) into the general polar coordinate formula \(r=\frac{l}{1+e\cos(\theta)}\) for a parabola, to get \(r=\frac{4}{1+1\cos(\theta +90)}\) which simplifies to \(r = \frac{4}{1+ \cos(\theta + 90°)}\).

Step 4: Simplify The Equation

Now, use trigonometric identities to simplify the equation. The cosine of an angle plus 90° equals the negative sine of the same angle. Hence, \(r=\frac{4}{1 -\sin(\theta)} =\frac{4}{2-\sin(\theta) - 1} = \frac{4}{2-(1 - r)}\) After organizing and simplifying, it is found the polar equation of the conic is \(r = 4(1 - r)\).

Key Concepts

Eccentricity in Conic Sections

Eccentricity is a fundamental concept in understanding conic sections, including ellipses, parabolas, and hyperbolas.
It is denoted by the symbol \( e \).
Eccentricity describes how much a conic section deviates from being a circle.

• A circle has an eccentricity of 0, meaning it's perfectly round.
• An ellipse has eccentricity between 0 and 1, making it slightly elongated.
• A parabola has an eccentricity exactly equal to 1, causing it to be perfectly u-shaped.
• A hyperbola's eccentricity is greater than 1, indicating very open curves.

For parabolas, which is our focus here, the eccentricity is precisely \( e = 1 \).
This means that every point on the parabola is equidistant from a single point called the focus and a line called the directrix.
This unique property makes parabolas especially interesting in both theoretical and applied mathematics.

The Nature of Parabolas

Parabolas are one of the most fascinating shapes in geometry and appear in many natural and human-made forms.
In polar coordinates, parabolas have a fixed point known as the focus, and a line known as the directrix.

• Every point on a parabola is equally distant from the focus and the directrix.
• Parabolas can open upwards, downwards, leftwards, or rightwards depending on their orientation.
• In the context of polar equations, when eccentricity \( e = 1 \), the conic section is a parabola.

Their reflective property is used in real-life applications such as satellite dishes and car headlights, where they focus signals or light precisely from a point source.

Understanding the Directrix

The directrix is a crucial part of defining a parabola.
It is a fixed line used in combination with the focus to outline the geometry of the conic.

• The distance from any point on the parabola to the focus is equal to its perpendicular distance to the directrix.
• In polar coordinates, the directrix can be represented as a line, such as \( y = -4 \), playing a key role in the parabola's equation.

In the given problem, the directrix is \( y = -4 \), indicating that it lies below the pole (origin) by a distance of 4 units.
This line helps you understand the orientation and positioning of the parabola in polar form, providing a basis to form the polar equation.
Recognizing how the directrix interacts with other elements of the parabola is essential for solving polar coordinate problems effectively.