Problem 70 - College Algebra

Question

For the following exercises, 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 \(y=3 x-9\) 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 \(y=-3 x+9\).

Step-by-Step Solution

Answer

The equation of the hyperbola is \( x^2 - \frac{y^2}{9} = 1 \).

Step 1: Determine the Type of Conic Section

Given that the object's path can be approximated by two intersecting lines and considering the sun as one of the foci, the object's path will be a hyperbola. Hyperbolas have two branches that open away from each other.

Step 2: Define the Standard Equation of a Hyperbola

The equation of a hyperbola centered at the origin has the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) if it opens horizontally (along the x-axis). Since the x-axis is the axis of symmetry, our hyperbola opens horizontally.

Step 3: Find the Slopes of Asymptotes

The given lines \( y = 3x - 9 \) and \( y = -3x + 9 \) represent asymptotes of the hyperbola. The slopes of these lines are \( 3 \) and \( -3 \), respectively. This means \( \frac{b}{a} = 3 \).

Step 4: Relate Asymptote Slopes to Hyperbola Equation

The relationship \( \frac{b}{a} = 3 \) implies \( b = 3a \). By substituting this expression into the hyperbola equation, \( \frac{x^2}{a^2} - \frac{y^2}{(3a)^2} = 1 \) which simplifies to \( \frac{x^2}{a^2} - \frac{y^2}{9a^2} = 1 \).

Step 5: Determine the Closest Approach to the Sun

The closest approach distance is 1 AU, which gives \( a = 1 \). Substituting \( a = 1 \) into the hyperbola equation yields \( \frac{x^2}{1^2} - \frac{y^2}{9} = 1 \), simplifying to \( x^2 - \frac{y^2}{9} = 1 \).

Step 6: Write the Equation of the Hyperbola

The final equation of the hyperbola, considering all determined parameters, is \( x^2 - \frac{y^2}{9} = 1 \).

Key Concepts

Conic Sections

Conic sections are curves obtained by slicing a cone at different angles. They include circles, ellipses, parabolas, and hyperbolas. Each type of conic section has unique properties and equations.
For a hyperbola, two branches open away from each other, often resembling two mirrored parabolas. In this exercise, we deal with a hyperbola, which is determined by the object's path when it enters and exits our solar system along two distinct lines.
This motion describes a hyperbola as it involves the intersection of two asymptotic lines, which are traditionally how hyperbolas are constructed in geometry. The presence of a focus, which is one attribute of all conic sections, further defines the nature of the hyperbola.

Coordinate System

A coordinate system is a framework used to define the positions of points in a plane or space. Most commonly, in problems like these, we use Cartesian coordinates, consisting of an x-axis and a y-axis that intersect at the origin (0,0).
In this specific scenario, the solar system's path has been plotted on a coordinate plane with the sun positioned at the center, or origin, and the x-axis acting as the central axis of symmetry. This means the hyperbola is centered at the origin, making the standard form of the hyperbola applicable.
Such a setup allows for a clear understanding of the path traveled by the object, using mathematical equations grounded in the Cartesian plane, ensuring accurate plotting and analysis of celestial paths.

Asymptotes

Asymptotes are lines that a curve approaches as it heads towards infinity. They are crucial in the study of hyperbolas because they define the direction that each branch of the hyperbola extends.
In this exercise, the given lines, expressed as \( y = 3x - 9 \) and \( y = -3x + 9 \), are asymptotes of the hyperbola. These lines do not intersect the hyperbola but set bound directions for the branches of the curve.
This gives us important information about the relationship between variables in the hyperbola's equation. Specifically, knowing the slopes helps us understand the ratio between the semi-major and semi-minor axes of the hyperbola and ensures the accuracy of the equation.

Focus of Hyperbola

The focus (plural: foci) of a hyperbola are points from which the difference in distances to any point on the hyperbola is constant. For hyperbolas, these are key in defining its shape.
In this exercise, one focus is located at the sun's position. This celestial application of a hyperbola uses the sun as a gravitational anchor point, given its role as a focus. The paths approach close to the sun but deviate along the hyperbolic shape.
Understanding the focus' placement helps in modeling the object's trajectory accurately, describing not just movement but also how close celestial bodies and objects come to massive gravitational sources.