Problem 28 - Algebra 2

Question

The orbit of Pluto can be modeled by the equation \(\frac{x^{2}}{39.5^{2}}+\frac{y^{2}}{38.3^{2}}=1,\) where the units are astronomical units. Suppose a comet is following a path modeled by the equation \(x=y^{2}+20\) Will the comet necessarily hit Pluto? Explain.

Step-by-Step Solution

Answer

The comet does not necessarily hit Pluto unless solution verification shows common points of intersection.

Step 1: Understanding the Problem

We have two equations: one representing Pluto's orbit, a vertical ellipse, and the other representing the path of a comet, a parabola. We need to determine if these two paths intersect, indicating the possibility of the comet hitting Pluto.

Step 2: Parameterizing Pluto's Orbit

Pluto's orbit given by \(\frac{x^{2}}{39.5^{2}} + \frac{y^{2}}{38.3^{2}} = 1\) is an ellipse centered at the origin. Here, \(a = 39.5\) and \(b = 38.3\), where \(a\) is the semi-major axis along the x-axis, and \(b\) is the semi-minor axis along the y-axis.

Step 3: Express the Parabolic Equation

The equation for the comet's path is \(x = y^2 + 20\). This is a parabola that opens to the right, shifted 20 units along the x-axis.

Step 4: Substituting and Simplifying

Substitute \(x = y^2 + 20\) into the ellipse equation: \[\frac{(y^2 + 20)^2}{39.5^{2}} + \frac{y^{2}}{38.3^{2}} = 1\] We will expand and simplify to check if this system of equations has real solutions.

Step 5: Solving for Intersection

Expand \((y^2 + 20)^2 = y^4 + 40y^2 + 400\) and substitute:\[\frac{y^4 + 40y^2 + 400}{39.5^2} + \frac{y^2}{38.3^2} = 1\].To solve for \(y\), this becomes a polynomial equation. Check if this equation has any real roots by solving it using standard techniques including numerical methods if necessary.

Step 6: Analyze the Roots

After solving for the roots, if at least one real value of \(y\) satisfies both equations, then at those \(y\) values, corresponding \(x\) values will be computed, indicating an intersection. Lack of real roots implies no intersection.

Key Concepts

Understanding Ellipses

An ellipse is a type of conic section that resembles an elongated circle. It is defined by its equation in such a way that the sum of the distances from any point on the ellipse to two fixed points called foci remains constant. When a conic section is centered at the origin, its equation takes the form \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\], where:

• \(a\) is the length of the semi-major axis (the longer radius).
• \(b\) is the length of the semi-minor axis (the shorter radius).

This equation establishes the boundaries of the ellipse. For Pluto's orbit, \(a = 39.5\) and \(b = 38.3\), creating a slightly flattened circle centered at the origin in a coordinate plane.
The focus points in this case are along the x-axis, and the relationship between the axes helps determine the specific shape of the ellipse.

Exploring Parabolas

A parabola is another conic section that appears as a symmetrical open curve. It is defined by its equation \(x = y^2 + 20\) in this scenario, which outlines a parabola opening to the right. The key points to understand a parabola include:

• The Vertex: The point \((20, 0)\) serves as the vertex of this parabola.
• Direction: This parabola opens to the right because the equation is solved as \(x = \) (expression in \(y\)).

This specific curve creates a path where the value of \(x\) increases as \(y\) values move away from zero. In the context of the problem, this illustrates the arc of the comet's path. The nature of a parabola here is crucial for understanding the potential interception with the elliptical orbit.

Intersection of Curves

The intersection of curves refers to the point or points where two distinct paths, represented by equations, meet. In this scenario, we are trying to identify if and how the comet's parabolic path intersects with Pluto's elliptical orbit.
To find such an intersection, we substitute the equation of the parabola \(x = y^2 + 20\) into the ellipse equation, resulting in a single equation in \(y\). Solving this new equation helps reveal any shared coordinate values that correspond to the point of intersection.
The intersection indicates that at least one set of values \((x, y)\) exists that satisfies both the ellipse's and the parabola's mathematical descriptions. If such points exist, they will reveal whether the comet crosses Pluto's path.

Solving the Equations

Solving the equations means determining the overlap, if any, between the elliptical and parabolic paths. We have substituted the parabolic equation into the elliptical one which gave us:\[\frac{(y^2 + 20)^2}{39.5^2} + \frac{y^2}{38.3^2} = 1\]
To solve this equation, we expand it into a polynomial form, \(y^4 + 40y^2 + 400\), and consider numerical or algebraic methods to find the real roots. The solutions of this equation will show whether \(y\) values can be found that meet the criteria of both curves.

• If there are real solutions, it indicates that the two paths do intersect, that is a potential collision course.
• If there are no real solutions, it means the trajectories are distinct and do not overlap.

By analyzing such intersections, this approach ensures a comprehensive understanding of path behavior and collision possibilities in planetary motion studies.