Problem 67 - 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 \(y=2 x-2\) and passes within 0.5 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 \(y=-2 x+2\)

Step-by-Step Solution

Answer

The equation of the hyperbola is \(4x^2 - y^2 = 1\).

Step 1: Identify the Transverse Axis

The hyperbolic path has the x-axis as its transverse axis, meaning the standard form of the hyperbola is horizontal. Since the x-axis is the line of symmetry for the object's path, we can start with the equation of a horizontal hyperbola.

Step 2: Determine the Closest Approach

The closest distance from the hyperbola’s edge to the focus at the Sun is 0.5 au, and the Sun is at the origin. This distance represents the distance from the vertex to the center along the transverse axis, denoted as \(a = 0.5\).

Step 3: Use Given Lines to Determine Asymptotes

The object's entry and exit paths approximate the asymptotes of the hyperbola. The given asymptotes are \(y = 2x - 2\) and \(y = -2x + 2\), both can be expressed in the form \(y = \pm mx\) for asymptotes, with \(m = 2\).

Step 4: Incorporate Information into the Standard Equation

Now incorporate all these into the standard form equation of a hyperbola centered at the origin: \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]Here, \(a = 0.5\). The slope of the asymptotes, 2, gives \(b/a = m\), so \(b = ma = 1\). Substitute these values into the equation.

Step 5: Substitute Values into the Hyperbola Equation

Substitute \(a = 0.5\) and \(b = 1\) into the equation:\[\frac{x^2}{0.25} - \frac{y^2}{1} = 1\]Simplify to get the final hyperbolic equation: \[4x^2 - y^2 = 1\]

Step 6: Verify the Solution

Verify the solution by checking that the asymptotes derived from the hyperbola \(4x^2 - y^2 = 1\) match \(y = \pm 2x\), ensuring the condition given in the exercise is met.

Key Concepts

Coordinate Systems

In the study of hyperbolas within algebra, coordinate systems play a significant role. These systems allow us to map and understand the paths of objects in space, such as in our solar system. The most common coordinate system used is the Cartesian coordinate system. It consists of an x-axis (horizontal), a y-axis (vertical), and sometimes a z-axis for three-dimensional spaces.

In the given exercise, the solar system serves as our canvas. We put the Sun at the origin of this system, meaning it is located at point (0,0). The x-axis is the line of symmetry for the hyperbola's transverse axis. This setup allows us to easily visualize and calculate the path of the object entering and exiting the solar system.

By understanding how to position objects and axes in a coordinate system, we can accurately model and predict the movements of celestial bodies. This becomes especially useful in fields like astronomy and space exploration.

Equation of Path

The equation of a hyperbolic path is crucial in understanding the trajectory of an object in space. For hyperbolas, this path is often defined as an equation that reflects its shape and orientation in the coordinate plane.

Our task was to derive the path using the geometric parameters and constraints. The provided equation: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] represents the standard form of a hyperbola. Here, "a" and "b" are related to the distances along the transverse and conjugate axes respectively.

From the problem, we learn that the closest approach to the Sun is 0.5 au, hence, \(a = 0.5\). This value helps define the width of the hyperbola.
Importantly, we also use the provided lines to identify the asymptotes further outlining the object's path.

Asymptotes

Asymptotes are the lines that a hyperbola approaches but never actually touches. These lines guide the shape of the hyperbola and play a crucial role in determining its direction.

In this exercise, the lines describing the object's path on entry and exit are crucial. The asymptotes given are the lines \(y = 2x - 2\) and \(y = -2x + 2\). These equations reflect the slopes of the asymptotes in a simple form \(y = \pm mx\), where \(m = 2\).

The slope "m" directly relates to the equation of the hyperbola. If our slope is 2, the relationship \(\frac{b}{a} = 2\) assists in calculating the rest of our path equation. Here, we find \(b = 1\) when \(a = 0.5\). Such relationships among variables demonstrate the beautiful symmetry inherent in hyperbolic shapes.

Equation Verification

Verifying the solution is the final check to ensure accuracy. It guarantees that the derived equation accurately represents the defined path we intended.

In our case, the hyperbolic equation \(4x^2 - y^2 = 1\) is obtained. To verify, we need to check if the asymptotes calculated from this equation, \( y = \pm 2x\), match the given asymptotes from the problem.
By confirming the derived asymptotes correspond with those provided, we confirm both the correctness of our path equation and adherence to the problem constraints.

This step solidifies our solution and provides confidence that our mathematical model is both reliable and correctly describes the object's path through our solar system.