Problem 71 - Intermediate Algebra

Question

A planet's orbit about the sun can be described as an ellipse. Consider the sun as the origin of a rectangular coordinate system. Suppose that the \(x\) -intercepts of the elliptical path of the planet are \(\pm 130,000,000\) and that the \(y\) -intercepts are \(\pm 125,000,000 .\) Write the equation of the elliptical path of the planet.

Step-by-Step Solution

Answer

The equation is \(\frac{x^2}{16,900,000,000,000,000} + \frac{y^2}{15,625,000,000,000,000} = 1\).

Step 1: Identify the Major and Minor Axes

In an ellipse, the axes are determined by the intercepts. The given intercepts are compared to identify which is larger. Here, the x-intercepts are \(\pm 130,000,000\) and the y-intercepts are \(\pm 125,000,000\), so the major axis is along the x-axis.

Step 2: Determine Semi-Major and Semi-Minor Axes

The semi-major axis \(a\) is half of the length of the major axis. In this case, \(a = 130,000,000\). The semi-minor axis \(b\) is half of the length of the minor axis: \(b = 125,000,000\).

Step 3: Write the Standard Form of the Ellipse Equation

The equation of an ellipse centered at the origin with horizontal major axis is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Substituting \(a\) and \(b\) into the equation, we have: \(\frac{x^2}{(130,000,000)^2} + \frac{y^2}{(125,000,000)^2} = 1\).

Step 4: Simplify the Equation

Calculate \((130,000,000)^2\) and \((125,000,000)^2\) to simplify the equation: \[(130,000,000)^2 = 16,900,000,000,000,000,\] and \((125,000,000)^2 = 15,625,000,000,000,000\). Substitute back to get the equation: \(\frac{x^2}{16,900,000,000,000,000} + \frac{y^2}{15,625,000,000,000,000} = 1\).

Key Concepts

Major Axis

The major axis of an ellipse is the longest diameter. It runs through the center and foci of the ellipse, extending to its widest points. In the rectangular coordinate system, you can identify the major axis by locating the axis with the larger intercepts. For the planet's orbit, the intercepts on the x-axis were \( \pm 130,000,000 \), and the y-axis intercepts were \( \pm 125,000,000 \). Since 130,000,000 is greater than 125,000,000, the major axis lies along the x-axis.
The major axis helps determine the shape of the ellipse and the length of this axis is crucial in finding the semi-major axis.

Minor Axis

The minor axis is the shortest diameter running through the center of an ellipse, perpendicular to the major axis. It connects the minimum width points of the ellipse. In this exercise, the minor axis is along the y-axis with intercepts \( \pm 125,000,000 \).
The minor axis is essential for shaping the ellipse and allows us to calculate the semi-minor axis, which further helps in formulating the ellipse equation.

Rectangular Coordinate System

A rectangular coordinate system, also known as the Cartesian coordinate system, is used to define locations in a plane using two perpendicular number lines: the x-axis and the y-axis.
In the context of an ellipse, this system helps in positioning and understanding the shape and size of the ellipse based on its intercepts. This exercise uses a rectangular coordinate system with the sun at the origin to define the elliptical path of the planet's orbit, making it easier to relate the x and y intercepts to the axes of the ellipse.

Semi-Major Axis

The semi-major axis is half the length of the major axis. It's a critical value that determines the horizontal or vertical stretch of an ellipse. For the given elliptical path, the semi-major axis is half of 130,000,000, which calculates to 130,000,000 since it's already noted as half due to the nature of the intercepts.
This value of the semi-major axis is then used squared in the ellipse equation in the format \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

Semi-Minor Axis

The semi-minor axis is half the length of the minor axis. It's a crucial metric that determines the other dimension of the ellipse, representing its vertical or horizontal breadth depending on orientation.
For our planet's elliptical orbit, the semi-minor axis is 125,000,000, derived as half of the minor intercepts. It is a key component in the standard form of the ellipse equation, where it appears squared in the \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) formula, with \( b = 125,000,000 \). This helps achieve the balance in calculating the elliptical path precisely.