Problem 55 - College Algebra

Question

A planet's orbit follows a path described by \(16 x^{2}+4 y^{2}-64\) A comet follows the parabolic path \(y-x^{2}-4 .\) Where might the comet intersect the orbiting planet?

Step-by-Step Solution

Answer

The points of intersection are (0,4) and (\(2\sqrt{3},16\)), (\(-2\sqrt{3},16\)).

Step 1: Rewrite the equation of the parabolic path

The parabolic path is described by the equation \(y = x^{2}+4\).

Step 2: Substitute the equation of y into the elliptic path

Substitute the equation from Step 1 into the elliptic path equation \(16x^{2}+4y^{2}=64\). So it becomes \(16x^{2}+4(x^{2} + 4)^{2}=64\).

Step 3: Simplify the equation

Simplify the equation from Step 2 and solve for x. The equation simplifies to \(16x^{2} + 4x^{4} + 32x^{2} + 64 = 64\), which further simplifies to \(4x^{4} + 48x^{2} = 0\). Divide by 4, to get \(x^{4} + 12x^{2} = 0\). The roots are \(x=0\) or \(x=\pm 2\sqrt{3}\).

Step 4: Solve for y

Substitute the values found for x into the equation of y from Step 1 to find the corresponding y values. For \(x=0\), \(y=0^{2}+4=4\) and for \(x=\pm2\sqrt{3}\), \(y=(\pm 2\sqrt{3})^{2} + 4 = 16\). So, the points of intersection are (0,4) and (\(2\sqrt{3},16\)), (\(-2\sqrt{3},16\)).

Key Concepts

Algebraic Intersections

Understanding where algebraic curves intersect is fundamental when tackling problems involving trajectories, such as discerning the collision points of a comet and a planet's orbit. In our exercise, the paths of a planet and a comet are each defined by an equation, depicting their respective paths through space.

In the realm of algebra, determining the intersection points of these paths equates to solving a system of equations. Here, the system is comprised of a quadratic function and an elliptic equation. To find the algebraic intersections, we substitute one equation into the other and solve for the possible values of the variables involved. In this case, we replaced the variable 'y' from the comet's parabolic path equation into the planet's elliptic path equation, resulting in a new equation that represents potential points of intersection.

Assessing the Intersection Points

After simplification, we are left with roots for the variable 'x', which correspond to the 'x' coordinates where the comet and planet's paths would intersect. Upon finding the 'x' values, the corresponding 'y' values are determined by reapplying them into the original quadratic equation. Thus, the intersection points acquired provide the exact locations where both the comet and planet would converge on their cosmic journeys.

Quadratic Functions

Quadratic functions, fundamental components of algebra, follow the general form of \(f(x) = ax^{2} + bx + c\), where \(aeq0\). These functions graph as parabolas, which can open upwards or downwards depending on the sign of the coefficient 'a'.

Our exercise presents a comet following a parabolic path, represented by the quadratic function \(y = x^{2}+4\). This parabola opens upwards, as indicated by the positive coefficient of the \(x^{2}\) term. In analyzing the trajectory of the comet, it is crucial to consider the prominent features of its quadratic path, which include the vertex, the direction of the parabola's opening, and the axis of symmetry.

Features of the Parabolic Path

The vertex of this parabola is the point (0,4), and it serves as the minimum point since the trajectory opens upwards. This parabola is symmetric around the y-axis, which is also its axis of symmetry. Recognizing these characteristics aids us in understanding the comet's journey and predicting potential points of interaction with other celestial objects, like the intersecting orbit of a planet.

Elliptic Paths

An elliptic path can be visualized as a stretched out circle - more formally described as an ellipse, which is a type of conic section. In algebra, the standard form of an ellipse's equation is \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\), where 'a' and 'b' are the lengths of the semi-major and semi-minor axes, respectively.

In our exercise, the planet follows an elliptic path expressed by the equation \(16x^{2} + 4y^{2} = 64\). Once this is rearranged to \(\frac{x^{2}}{4} + \frac{y^{2}}{16} = 1\), we can observe that the lengths of the semi-major and semi-minor axes are 4 and 2, respectively. Unlike the straightforward trajectory of a quadratic function, an elliptic path encapsulates a range of motion that is bounded within its curved perimeter.

Characteristics of the Elliptic Orbit

The ellipse has two axes of symmetry, the major and minor axes. In celestial mechanics, elliptic orbits are common and result from the gravitational forces exerted, as exemplified by many planets in our solar system. Familiarity with the properties of an elliptic path is crucial when predicting the movement of celestial bodies and the possible points where they may meet or even eclipse each other.