Problem 26 - 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. \(9x^2 +24xy+16y^2+80x-60y = 0\)

Step-by-Step Solution

Answer

The angle of rotation, \(\theta\), is 45 degrees. After applying the rotation matrix to the equation, it simplifies to a standard form: \((x' - 5/9)^2/9^2 + (y' - 15/8)^2/16^2 = 1\). The graph of the equation is an ellipse, centered at \((5, 15/8)\) with a major axis length of 18 (along \(x'\)) and a minor axis length of 32 (along \(y'\)). The major axis is horizontal and the minor axis is vertical.

Step 1: Calculate the Angle of Rotation

Calculate the angle of rotation, \(\theta\), using the formula \(\theta = 0.5 \cdot atan(2b / (a - d))\), where \(a = 9\), \(b = 12\), and \(d = 16\). This gives us \(\theta = 0.5 \cdot atan(24/(9-16)) = atan(1) = 45^\circ\).

Step 2: Apply Rotation Matrix

Using the rotation matrix \(\begin{bmatrix} cos(\theta) & -sin(\theta) \ sin(\theta) & cos(\theta) \end{bmatrix}\), Apply it to the x and y of the original equation. Remember to replace \(x = x'cos(\theta) - y'sin(\theta)\) and \(y = y'cos(\theta) + x'sin(\theta)\), where \(x'\) and \(y'\) are the points on the rotated axes. This gives us a new equation: \(9x'^2 + 16y'^2 - 40x' + 60y' = 0\).

Step 3: Simplify Equation into Standard Form

Simplify the equation into the standard form of a conic section equation, which is : \((x - h)^2/a^2 + (y - k)^2/b^2 = 1\). To achieve this, complete the square for both x and y terms in the equation, which gives us: \((x - 45/9)^2/9^2 + (y - 15/8)^2/16^2 = 1\). This implies that the center (h,k) of the conic section is at \((5, 15/8)\).

Step 4: Sketch the Graph

Sketch the graph of the equation. The center of the ellipse is at the point (5, 15/8), \(a = 9\) and \(b = 16\). The major axis has a length of 2a and is horizontal, and the minor axis has a length of 2b and is vertical. Finally, denote the rotated axes. Noting that the original axes were rotated 45 degrees to obtain the new axes.

Key Concepts

Angle of Rotation

The angle of rotation is a crucial concept that helps us transform equations of conic sections into a simpler form. When we have an equation with an

• xy-term, it signifies that the conic section is rotated with respect to its conventional position.
• To eliminate this term, we need to calculate the angle of rotation, \(\theta\). We use the formula \(\theta = 0.5 \cdot \text{atan}\left(\frac{2b}{a - d}\right)\), where \(a\), \(b\), and \(d\) are coefficients from our equation.
• In our example, substituting the values gives \(\theta = 45^\circ\).

Calculating this angle helps us apply the correct transformations to simplify the equation. By rotating the axes by this angle, we can align the conic section to make it easier to analyze and graph. Keep in mind that knowing how to rotate axes is a powerful tool in handling complex conic section equations.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. These sections include ellipses, parabolas, and hyperbolas. Each type of conic section has its own unique properties and equation forms depending on the

• orientation
• relative position of the intersecting plane.
• When dealing with equations that include an \(xy\) term, it's often the case that we're working with rotated conic sections.

To analyze them in their standard form, we first need to perform a transformation to eliminate the \(xy\) term. This process simplifies the equation, allowing us to identify the type of conic section more easily. Conic sections play a significant role in mathematics due to their practical applications in numerous fields such as physics, engineering, and architecture.

Ellipses

Ellipses are one of the conic sections characterized by two axes of symmetry - the major and minor axes. They resemble stretched circles and have several properties that distinguish them from other conic sections.

• An ellipse is described by the equation \((x - h)^2/a^2 + (y - k)^2/b^2 = 1\),
• where \((h, k)\) represents the center, and \(a\) and \(b\) are the semi-major and semi-minor axes, respectively.
• In our example, after completing the square, we derived the standard form equation of an ellipse, showing the center at \((5, 15/8)\).

The axis lengths determine the shape of the ellipse. Knowing the standard form makes it much simpler to sketch ellipses and understand their properties in a given geometric context.

Completing the Square

Completing the square is a mathematical technique used to convert quadratic equations into a perfect square trinomial. This method is invaluable when transforming equations from a general to a standard form, especially for conic sections.

• This approach involves rearranging the terms of a quadratic expression so it can be expressed as \((x - p)^2 + C\), which is easier to graph and analyze.
• In the context of our ellipse problem, completing the square was applied to both \(x\) and \(y\) terms. This allowed us to convert the original rotated ellipse equation into its standard form.
• The end result provided us with a more readable equation that highlighted the ellipse's center and axis lengths, simplifying the sketching and further analysis of the graph.

Mastering this technique is essential for handling a variety of algebraic problems and provides a systematic way to manipulate complex mathematical expressions.