Problem 13 - Precalculus with Limits

Question

In Exercises 13-26, rotate the axes to eliminate the \(xy\)-term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. \(xy + 1 = 0\)

Step-by-Step Solution

Answer

For the equation \(xy + 1 = 0\), the rotation angle \(\Theta\) is \(\pi/4\). After applying transformation and rotation of axes, the equation transforms to \(x'y' = -1\). Now it's ready to graph.

Step 1: Find the Rotation Angle

The rotation angle \(\theta\) can be found using the formula:\(\theta = 0.5 \cdot \arctan{(2b / (a - d))}\)In our case, \(a = d = 0\) and \(b = 0.5\), so:\(\theta = 0.5 \cdot \arctan{(2*0.5 / (0 - 0))} = \pi/4\)

Step 2: Rotate the Axes

The rotation of the axes is accomplished through the transformation of coordinates from \(x, y\) to \(x', y'\). These are related via:\(x = x' \cos{\theta} - y' \sin{\theta}\),\(y = x' \sin{\theta} + y' \cos{\theta}\)Substituting \(\theta = \pi/4\), we get:\(x = x' / \sqrt{2} - y' / \sqrt{2}\),\(y = x' / \sqrt{2} + y' / \sqrt{2}\)

Step 3: Substitute the Rotated Axes into the Equation

Substitute x and y into the original equation:\(x'y' + 1 = 0\)

Step 4: Rewrite the Equation in Standard Form

The equation \(x'y' + 1 = 0\) can be rewritten in standard form as:\(x'y' = -1\)

Step 5: Graph the Equation

Now it’s time to graph this equation. Here, \(x' = x/ \sqrt{2} - y / \sqrt{2}\), and \(y' = x / \sqrt{2} + y / \sqrt{2}\). To graph, pick values for \(x'\) and solve for corresponding \(y'\) values. Then convert \(x'\) and \(y'\) to original \(x\) and \(y\) values. Remember to graph both original axes and new rotated axes.

Key Concepts

Rotation of Axes

The rotation of axes is a crucial technique used in the transformation of coordinate systems. This method is typically employed to simplify the equations of conic sections by removing the cross-product term, namely the \(xy\)-term. To achieve this, a rotation angle \(\theta\) is calculated using the formula:

• \[ \theta = \frac{1}{2} \tan^{-1}\left(\frac{2b}{a - d}\right) \]

In the exercise, since the equation \(xy + 1 = 0\) lacks terms with \(x^2\) or \(y^2\), both \(a\) and \(d\) are zero. Thus, when finding \(\theta\), the calculation simplifies, and in this case, the rotation angle results in \(\theta = \frac{\pi}{4}\). This rotation is key to eliminating the \(xy\)-term, converting the original system to one where the graphing process becomes simpler. Once \(\theta\) is found, it's used to transform the original coordinates \((x, y)\) into new rotated coordinates \((x', y')\) using:

• \(x = x' \cos{\theta} - y' \sin{\theta} \)
• \(y = x' \sin{\theta} + y' \cos{\theta} \)

Substituting \(\theta = \frac{\pi}{4}\) converts these expressions into:

• \(x = \frac{x' - y'}{\sqrt{2}} \)
• \(y = \frac{x' + y'}{\sqrt{2}} \)

Standard Form of Equations

The standard form of an equation for conic sections is essential for identifying and graphing them easily. By converting an equation into its standard form, we can easily recognize whether it's a circle, ellipse, parabola, or hyperbola. This process involves the simplification and elimination of certain terms, primarily the \(xy\)-term, using transformations or rotations.For the equation \(xy + 1 = 0\), after applying the rotation of axes, the equation is transformed into the form \(x'y' + 1 = 0\). Upon further simplification, by rearranging terms, it can be rewritten in the standard form \(x'y' = -1\).Standard forms make the characteristics of the conic evident, like symmetry, vertices, or center, which are vital for accurate graphing.

Graphing Conics

Once the equation is in standard form, graphing conic sections becomes straightforward. For conics in forms like \(x'y' = -1\), the graph typically represents a hyperbola. From the solution, we see that the transformation and resulting equation illustrate a hyperbola in the new \((x', y')\)-axes.To graph the conic:

• Choose values for \(x'\).
• Solve for corresponding \(y'\) values, as suggested by the equation \(x'y' = -1\).
• Convert \(x'\) and \(y'\) back to \(x\) and \(y\) using the inverse of the original transformation.

This process allows you to plot points that make up the conic. It is important to draw both the original and rotated axes on the graph to understand how the coordinate rotation affects the conic's orientation. The rotated axes help illustrate how the hyperbola aligns with them, providing insights into its elliptical and hyperbolic nature.