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Magazine Chapter 8: Problem 54
Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(y=2 ; e=2.5\)
Short Answer
Expert verified
The polar equation is \( r = \frac{5}{1 + 2.5 \sin \theta} \).
Step by step solution
01
Understanding the Problem
We need to find the polar equation of a conic with given eccentricity (\(e=2.5\)) and directrix (\(y=2\)). The focus of the conic is at the origin.
02
Identifying the Type of Conic
Since the eccentricity \(e = 2.5\) is greater than 1, the conic is a hyperbola.
03
Standard Form for Polar Equations of Conics
The general form of the polar equation for a conic is \(r = \frac{ed}{1 + e\sin\theta}\) if the directrix is horizontal (parallel to the x-axis), and \(r = \frac{ed}{1 + e \cos \theta}\) if the directrix is vertical.
04
Converting the Directrix to Polar Form
Since the directrix is \(y = 2\), which is horizontal, we use the polar form equation \(r = \frac{ed}{1 + e\sin\theta}\). Here the distance \(d\) from the focus (origin) to the directrix is 2.
05
Substituting Values into the Equation
Substitute \(e = 2.5\) and \(d = 2\) into the equation for a horizontal directrix: \[ r = \frac{2.5 \times 2}{1 + 2.5 \sin \theta} \]Simplifying, we get\[ r = \frac{5}{1 + 2.5 \sin \theta} \].
06
Final Polar Equation of the Conic
The polar equation of the conic with the given parameters is \[ r = \frac{5}{1 + 2.5 \sin \theta} \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Conic Sections
Conic sections are curves obtained by intersecting a cone with a plane. There are four primary types of conic sections: circles, ellipses, parabolas, and hyperbolas. Each type is determined by the angle at which the plane intersects the cone and the relative proximity of the intersecting plane to the cone's vertex. For example:
- A circle is formed when the plane intersects the cone parallel to its base.
- An ellipse is a stretched circle, formed when the intersection plane is tilted but does not reach the base.
- A parabola emerges when the plane is parallel to the side of the cone.
- A hyperbola occurs when the plane intersects both halves of the double cone.
Eccentricity
Eccentricity (denoted as \(e\)) measures how much a conic section deviates from being circular. It helps in determining the shape and type of conic sections. Here's how eccentricity defines the conic:
- For a circle, \(e = 0\).
- If \(0 < e < 1\), the conic is an ellipse.
- When \(e = 1\), it forms a parabola.
- If \(e > 1\), the conic is a hyperbola.
Directrix
A directrix in conic sections is a fixed line used in describing and defining the curve. It differs depending on the conic:
- In ellipses and parabolas, it serves as a reference line which helps in determining how points on the conic relate to the focus.
- For hyperbolas, like the one in the problem, it is pivotal as it helps determine the distance measurement crucial for the equation in polar form.
Focus
The focus is a point that plays a key role in defining a conic section. Every point on the conic is equidistant from the focus and another geometric feature (such as the directrix). Here's its role in different conics:
- For a circle and an ellipse, there are one or more foci which affect the shape.
- A parabola has one focus, which with the directrix, defines its characteristic curve.
- A hyperbola, like in this exercise, has two foci but only one is usually discussed in simple polar equations.
