Problem 20 - 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. $$7 x^{2}-6 \sqrt{3} x y+13 y^{2}-64=0$$

Step-by-Step Solution

Answer

The new equation is \(13x'^{2} + 7y'^{2} - 64=0\). For the plotting step, note that the new axis should be drawn rotated \(-\frac{5\pi}{6}\) relative to the original coordinate system and should be plotted in the same graph.

Step 1: Identifying the coefficients

We can identify that \(a = 7\), \(b = -6\sqrt{3}\), and \(c = 13\) in the given equation.

Step 2: Computing the rotation angle

The rotation angle \(\phi\) can be calculated by \(\phi = 0.5 \cdot arctan({B}/{A-C})\). Substituting \(A = 7\), \(B = -6\sqrt{3}\), and \(C = 13\) then \(\phi = 0.5 \cdot arctan({-6\sqrt{3}}/{7-13})\). After calculation, we get \(\phi = \frac{5\pi}{6}\).

Step 3: Finding the new equations for \(x'\) and \(y'\) after rotation

The axes can be rotated by using \(x = x'cos(\phi) - y'sin(\phi)\) and \(y = x'sin(\phi) + y'cos(\phi)\). Substitute \(\phi = \frac{5\pi}{6}\) into these two equations. After simplification, we get for \(x' = -x\frac{\sqrt{3}}{2} - y\frac{1}{2}\) and \(y' = -x\frac{1}{2} + y\frac{\sqrt{3}}{2}\)

Step 4: Convert the equation with new variables

Substitute \(x' = -x\frac{\sqrt{3}}{2} - y\frac{1}{2}\) and \(y' = -x\frac{1}{2} + y\frac{\sqrt{3}}{2}\) into the original equation. Expand it and combine terms. The new equation in standard form for \(x'\) and \(y'\) will be \(13x'^{2} + 7y'^{2} - 64=0\) after simplifying

Step 5: Sketch the graph

Plot the original equation and the standard equation in the same axis. Here, note that the new coordinate system should be drawn rotated \(-\frac{5\pi}{6}\) relative to the original coordinate system, sketch both original coordinate system and the new one on the graph.

Key Concepts

Eliminating XY Term

When dealing with conic sections, the presence of the \(xy\) term indicates that the conic might be rotated in its orientation. To simplify the analysis and better understand the conic, we'd like to eliminate this term. The key to achieving this is through the process of rotating axes.

• First, identify your original equation coefficients: \(a\), \(b\), and \(c\).
• The \(b\) coefficient is linked to the \(xy\) term. By rotating the axes, you aim to reorganize the equation such that \(b'\) (the new \(xy\) coefficient) becomes zero.
• To determine the rotation angle \(\phi\) utilized, use the formula: \(\phi = 0.5 \cdot \arctan\left(\dfrac{b}{a-c}\right)\).

By substituting the values from the original equation into this formula, we find the needed rotation angle to remove the \(xy\) term. The goal is to modify how the equation is framed, setting the stage for re-expressing it in a more recognized form.

Standard Form of Conics

After eliminating the \(xy\) term through axis rotation, our next objective is to express the equation in a standard form. Standard forms vary based on conic type, including circles, ellipses, parabolas, and hyperbolas. Simplifying the equation this way makes it easier to identify the type of conic and analyze its properties.

• A conic's standard form generally looks simpler, typically without any cross-product terms (like \(xy\)).
• The new coordinates \((x', y')\) are the result of substituting trigonometric identities involving \(\phi\) into the original variables.
• Rewriting the equation with these new coordinates, we simplify to ensure no \(xy\) terms remain and find sums of squared terms, constants, etc.

In this exercise, the equation simplifies to \(13x'^{2} + 7y'^{2} - 64 = 0\). This simplification not only indicates a conic section (in this case, an ellipse) but also aligns it to its principal axes for easier analysis.

Graph of Conic Sections

Graphing conic sections helps visualize the geometric relationship of the solutions. First, recognize the implications of the mathematical equation.

• Identify the resulting conic from your standard equation. For example, \( 13x'^{2} + 7y'^{2} - 64 = 0 \) reveals an ellipse.
• Plot the transformation from the original axis to the new rotated axis. This usually involves drawing both axes on the same plot.
• In the case of this exercise, the new coordinates align the ellipse in such a way that it's easy to determine characteristics like center, axes lengths, and orientation.

Visualizing both sets of axes provides insight into how the rotation affects the graph's shape and positioning, enhancing your understanding of conic section transformations.