Problem 69 - College Algebra

Question

assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information. The object enters along a path approximated by the line \(y=\frac{1}{3} x-1\) and passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line \(y=-\frac{1}{3} x+1 .\)

Step-by-Step Solution

Answer

The hyperbola's equation is \( x^2 - 9y^2 = 1 \).

Step 1: Understanding the Problem

The problem describes an object's path that is a hyperbolic trajectory with the Sun as one of the foci. The object approaches along a line and departs along a symmetrical line.

Step 2: Analyze the Approach and Departure Lines

The object approaches the solar system along the line \( y = \frac{1}{3}x - 1 \) and departs along the line \( y = -\frac{1}{3}x + 1 \). These lines indicate the paths the object follows before and after the closest approach.

Step 3: Determine the Hyperbola Orientation

The symmetry axis of the hyperbola is aligned with the x-axis, as given in the problem statement. The lines are symmetric across the x-axis, which further reinforces this direction.

Step 4: Identify Standard Hyperbola Equation Components

The general form of a hyperbola centered at the origin with the transverse axis along the x-axis is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). We need to find \(a\), \(b\), and the relationship between them given the provided lines.

Step 5: Closest Approach to the Sun

The closest approach to the Sun is 1 au, which is the distance from the center (focus) to the nearest vertex of the hyperbola. Thus, the distance from the origin (focus) to the vertex on the x-axis is \(1\ au\). Therefore, \(a = 1\).

Step 6: Calculating Asymptotes of the Hyperbola

The lines \( y = \frac{1}{3}x - 1 \) and \( y = -\frac{1}{3}x + 1 \) represent the asymptotes of the hyperbola. Asymptotes for a hyperbola with horizontal transverse axis take the form \( y = \pm \frac{b}{a}x \), thus \( \frac{b}{a} = \frac{1}{3} \).

Step 7: Solve for \(b\)

Given \( a = 1 \) and \( \frac{b}{a} = \frac{1}{3} \), we find \( b = \frac{1}{3} \cdot 1 = \frac{1}{3} \).

Step 8: Write the Equation of the Hyperbola

Substitute \( a = 1 \) and \( b = \frac{1}{3} \) into the standard equation: \[ \frac{x^2}{1^2} - \frac{y^2}{(\frac{1}{3})^2} = 1 \]. Simplifying this gives the equation \( x^2 - 9y^2 = 1 \).

Key Concepts

Distance from Origin

In the context of hyperbolas and their equations, the origin often holds special significance as it is a common point around which conic sections such as hyperbolas are centered. When dealing with hyperbolas, the origin frequently represents one of the foci, particularly when the conic is centered at this point.
For the hyperbola's equation given in the problem, which is centered at the origin, the hyperbola approaches the sun, our origin in this scenario, closely. By examining the problem, we know that the closest approach to the sun (origin) is 1 astronomical unit (au). This distance is equivalent to the value of 'a' in the standard hyperbola equation \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] This represents the length between the center and a vertex along the x-axis for a horizontally oriented hyperbola.
In summary, understanding the role of the origin in a hyperbola's path is crucial, especially when determining the hyperbola's specific parameters related to its focal distance.

Asymptotes

Asymptotes are crucial lines associated with hyperbolas, serving as a guide to the hyperbola's shape as it extends toward infinity. They are straight lines that the hyperbola approaches but never touches. In terms of graphing, asymptotes help sketch the general path and ensure accuracy in delineating the hyperbola's open branches.
According to the problem, the object's path is defined by two lines, \( y = \frac{1}{3}x - 1 \) and \( y = -\frac{1}{3}x + 1 \), which correspond to the asymptotes of the hyperbola. These asymptotes indicate the object arrives and departs in a mirrored fashion concerning the x-axis.

• For hyperbolas with a horizontal transverse axis, these asymptotes have the characteristic form: \( y = \pm \frac{b}{a}x \).
• Comparing it to the given lines, we find \( \frac{b}{a} = \frac{1}{3} \).

Thus, in these lines, the slope \( \pm \frac{1}{3} \) equates to \( \frac{b}{a} \), helping us compute \( b \) since we know \( a = 1 \).
This understanding is integral in applying and confirming the precise equation of the hyperbola.

Coordinate System

The coordinate system provides a framework for accurately plotting and analyzing an object's motion, especially when dealing with conic sections like hyperbolas.
In the problem, the object's path is interpreted using a Cartesian coordinate system, where the sun is positioned at the origin \((0,0)\), and the x-axis serves as the hyperbola's axis of symmetry. This setup facilitates the simplification of the hyperbola's equation and its alignment with the coordinate axes. By placing the hyperbola so that its transverse axis is along the x-axis, our focus primarily remains on computing the values associated with 'a' and 'b' for the standard form of the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
This coordinate system allows us to precisely depict the object's hyperbolic trajectory, aiding in visualization and ensuring accurate calculations and interpretations that conform to mathematical conventions for hyperbolas centered at the origin.

Conic Sections

Conic sections include several geometric figures that result from intersecting a plane with a cone, which features ellipses, parabolas, circles, and hyperbolas. Understanding hyperbolas within conic sections is essential, particularly in fields like astronomy and physics, where such trajectories often describe the paths of celestial objects.
The problem at hand delves into the use of hyperbolic pathways to model the trajectory of an object approaching and departing our solar system, emphasizing the hyperbola’s real-world applications.

• A hyperbola is distinguished by two separate curves, known as branches, and is especially relevant in scenarios involving objects moving along escape trajectories.
• The hyperbolic path is often based at common foci, such as the sun (origin) in our example, illustrating its role in determining paths deviating from a direct approach or departure.

Hyperbolas capture such dynamic movements under gravitational influences, with their mathematical representation \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]depending on precise identifying features like foci and asymptotes, ultimately bearing significant implications for predicting and understanding motion in space.