Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-32: Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$ e=1, \quad r \cos \theta=5 $$

Step-by-Step Solution

Answer

The polar equation is \(r = \frac{5}{1 + \cos(\theta)}\).

Step 1: Understand the Problem

We need a polar equation for a conic with a given eccentricity \(e\) and a directrix given by an equation. The eccentricity \(e=1\) indicates that the conic is a parabola.

Step 2: Identify Conic Type

Since the eccentricity \(e = 1\), the conic section we are dealing with is a parabola. Parabolas have a unique property that \(e = 1\) and they can open either upwards, downwards, left, right, or at some angle.

Step 3: Write the General Polar Equation

The polar form equation of a conic section with focus at the pole \((0,0)\) is \[r = \frac{ed}{1 + e\cos(\theta)}\] when the directrix is \(r \cos \theta = d\), provided the eccentricity \(e=1\).

Step 4: Substitute Known Values

The directrix given is \(r \cos \theta = 5\), so \(d=5\). Given \(e=1\), substitute these into the polar equation: \[r = \frac{5}{1 + \cos(\theta)}\].

Step 5: Simplify the Equation

The equation reduces to \(r = \frac{5}{1 + \cos(\theta)}\), which represents the polar equation of the parabola with the focus at the pole and the directrix \(r\cos\theta=5\).

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. Each conic section has unique geometric properties and can be defined by an equation.

• **Circle** - All points are equidistant from a central point.
• **Ellipse** - It resembles a stretched circle, with two focal points.
• **Parabola** - Each point is equidistant from a focus and a directrix.
• **Hyperbola** - Consists of two separate curves, each wrapping around a focus point.

Understanding conic sections is crucial for solving geometry and calculus problems, especially those involving polar coordinates. Conic sections have applications in physics, engineering, astronomy, and more.

Eccentricity

Eccentricity measures how "stretched" or "squished" a conic section is. This number helps classify the conic sections uniquely.

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

Eccentricity essentially describes the shape's deviation from being circular. For example, in our exercise, the given eccentricity is 1, signifying that we are dealing with a parabola.

Parabola

A parabola is a symmetrical open curve. It has several applications, particularly in mechanics, optics, and graphing quadratic functions. Every parabola has:

• A **focus**, a point which the parabola wraps around.
• A **vertex**, the turning point of the parabola closest to the directrix.
• A **directrix**, a line that helps define the parabola's openness and direction.

In polar coordinates, a parabola's equation is influenced by its eccentricity and directrix. With an eccentricity of 1, the equation simplifies as seen in the step-by-step solution: \[r = \frac{5}{1 + \cos(\theta)}\]where the focus is at the pole (origin) and the directrix is given by \(r \cos \theta = 5\).

Directrix

The directrix of a conic section is a straight line used with the focus to define and construct the curve. For a parabola, it guides its shape and direction.

• In our context, the directrix is given by **\(r \cos \theta = 5\)**. This defines a vertical line at a specific distance from the pole when equated.
• The parabolic curve maintains an equal distance from this directrix and the focus at every point.

The directrix, paired with the focus, makes sure the points adhere to the geometric definition of the parabola. By manipulating the directrix's position in relation to the pole, the parabola can assume various orientations and sizes.