Problem 19 - Precalculus

Question

Rotate the axes to eliminate the \(x y\) -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. $$13 x^{2}+6 \sqrt{3} x y+7 y^{2}-16=0$$

Step-by-Step Solution

Answer

After completing these steps, the equation should be in a standard form, looking something like: \(\frac{(X-h)^{2}}{a^{2}} + \frac{(Y-k)^{2}}{b^{2}} = 1\), and the graph should depict an ellipse drawn with its center at the origin, along with the old and new rotated axes.

Step 1: Proper Rotation of Coordinate Axes

The rotation of the axes formula is given by \(x = X \cos(\theta) - Y \sin(\theta)\) and \(y = X \sin(\theta) + Y \cos(\theta)\), where X and Y are the new coordinates, \(x\) and \(y\) are the old coordinates, and \(\theta\) is the rotation angle. For the given equation, since the coefficient of \(xy\) term is positive, we take \(\theta = - \frac{\pi}{6}\) considering \(\cot(2\theta)=\frac{a-b}{h}\), where \(a=13\), \(b=7\), and \(h=6\sqrt{3}\). Substitute \(\theta\) into the formulas and determine the new X and Y values.

Step 2: Substitute X and Y into the Equation

Substitute X and Y values from the above step into the given equation replacing \(x\) and \(y\). Simplifying the equation will lead to a quadratic equation in traditional form without an \(XY\) term.

Step 3: Write the Equation in Standard Form and Sketch the Graph

After simplifying the equation, it should represent an ellipse in a standard form. Rewrite the equation in standard form for an ellipse: \(\frac{(X-h)^{2}}{a^{2}} + \frac{(Y-k)^{2}}{b^{2}} = 1\), where (h, k) is the center of the ellipse, and \(a\) and \(b\) are the semi-major and semi-minor axes respectively. Then sketch the graph of the equation using the center, semi-major axes, and semi-minor axes. Use a different color to show the old axes and the new rotated axes on the graph.

Key Concepts

ellipse

An ellipse is a smooth, closed curve on a plane, and it is a specific type of conic section. Imagine stretching a circle in one direction; that's how an ellipse looks. Its appearance is defined by its two axes: the major axis and the minor axis. The major axis is the longest diameter across the ellipse, while the minor axis is the shortest.

In mathematical terms, an ellipse can be described by an equation of the form:

• \[ \frac{(X-h)^{2}}{a^{2}} + \frac{(Y-k)^{2}}{b^{2}} = 1 \]

Here,

• \((h, k)\) is the center of the ellipse.
• \(a\) is the distance from the center to the farthest end of the ellipse (semi-major axis).
• \(b\) is the distance from the center to the nearest end of the ellipse (semi-minor axis).

This equation shows the relationship between the coordinates on the plane and how they stretch along the axes. Once you gather the values of these variables, you can sketch an accurate representation of the ellipse. Understanding how to rotate and shift axes plays a crucial role in simplifying these equations to analyze the ellipse better.

coordinate transformation

Coordinate transformation is a process of converting points between different coordinate systems. This is useful in simplifying equations, especially to eliminate cross-product terms, like the troublesome \(xy\)-term in the initial quadratic equation.

To perform a coordinate transformation with a rotation of axes, use the following formulas:

• \(x = X \cos(\theta) - Y \sin(\theta)\)
• \(y = X \sin(\theta) + Y \cos(\theta)\)

Here, \((x, y)\) are the original coordinates, \((X, Y)\) are the new, rotated coordinates, and \(\theta\) is the angle of rotation. This transformation changes the orientation of the axes, making it possible to handle equations with cross-product terms more straightforwardly.

By carefully choosing the angle \(\theta\), based on the given conditions such as the coefficients in the equation, you can eliminate the \(xy\)-term. This makes the resulting calculations and graph plotting much simpler and more intuitive, allowing you to analyze the properties of the conic sections more effectively.

standard form of a conic section

The standard form of a conic section is an organized way of writing the equation of various curves like ellipses, circles, parabolas, and hyperbolas. For an ellipse, the equation is represented as:

• \[ \frac{(X-h)^{2}}{a^{2}} + \frac{(Y-k)^{2}}{b^{2}} = 1 \]

Having an equation in this standardized form is beneficial because it clearly lays out the geometric properties like center, axes, and orientation of the conic section. This standard form emerges after you apply transformations, such as rotation, to eliminate mixed terms—like the \(xy\)-term in the given example.

After transforming and simplifying the axes, the equation can often look like a basic ellipse equation, where:

• The center \((h, k)\) denotes the midpoint of the ellipse.
• \(a\) and \(b\) represent the lengths of the semi-major and semi-minor axes, respectively.

Converting an equation into its standard form is a crucial step in making it easy to graph the conic section and understand its geometric features. This understanding aids in visualizing the orientation and dimension of the conic section when analyzing problems or real-world applications.