Problem 20 - Precalculus with Calculus Previews

Question

In Problems \(17-20,\) use rotation of axes to eliminate the \(x y\) -term in the given equation. Identify the conic. $$ x^{2}-x y+y^{2}-4 x-4 y=20 $$

Step-by-Step Solution

Answer

The resulting conic is an ellipse.

Step 1: Identify the coefficient components

In the given equation \(x^2 - xy + y^2 - 4x - 4y = 20\), the quadratic terms are \(x^2\) and \(y^2\), with a mixed term of \(-xy\). The coefficients are thus: \(A = 1\), \(B = -1\), and \(C = 1\).

Step 2: Calculate the angle of rotation

To eliminate the \(xy\) term, use the formula \(\tan(2\theta) = \frac{B}{A-C}\). This gives \(\tan(2\theta) = \frac{-1}{1-1} = \frac{-1}{0}\), indicating an undefined result, which occurs when \(\theta = \frac{\pi}{4}\).

Step 3: Substitute for rotation transformation

The rotation formulas are \( x = x'\cos(\theta) - y'\sin(\theta) \) and \( y = x'\sin(\theta) + y'\cos(\theta) \). With \(\theta = \frac{\pi}{4}\), we find that \(\cos(\theta) = \frac{\sqrt{2}}{2}\) and \(\sin(\theta) = \frac{\sqrt{2}}{2}\). So, substituting these into the rotation formulas gives the substitutions: \( x = \frac{\sqrt{2}}{2}(x'-y') \) and \( y = \frac{\sqrt{2}}{2}(x'+y') \).

Step 4: Substitute and simplify the new equation

Substitute \(x\) and \(y\) from the rotation transformation into the original equation to eliminate the \(xy\) term. Simplify using algebraic identities until you obtain a new equation free of the \(xy\) term.

Step 5: Identify the conic

The resulting simplified equation represents a conic. By examining the coefficients of \(x'^2\) and \(y'^2\) in the new equation, you can identify whether the conic is a circle, ellipse, hyperbola or parabola. The values suggest an ellipse.

Key Concepts

Elimination of xy-term

When dealing with conic sections in equations, sometimes you will have a mixed term of the form \(xy\) that can complicate the analysis. This term represents an interaction between the x and y directions and needs to be eliminated for easy identification of the type of conic section. To remove this \(xy\) term, we perform an operation called rotation of axes. The idea is to rotate the coordinate system itself around the origin to align with the principal axes of the conic. By doing this, the \(xy\) term disappears, simplifying the equation of the conic. Here's how you go about it:

• Identify the coefficient \(B\) of the \(xy\) term and the coefficients \(A\) and \(C\) of \(x^2\) and \(y^2\), respectively.
• Compute \( \tan(2\theta) = \frac{B}{A-C} \) to find the angle \(\theta\) for rotation. This formula is derived from applying a rotation transformation that results in zeroing out the \(xy\) term.

Conic Sections

Conics are the curves obtained by intersecting a plane with a double-napped cone. They come in four principal shapes: circles, ellipses, parabolas, and hyperbolas. Understanding the type of conic section represented in an equation involves examining its quadratic terms and coefficients. Once the \(xy\) term is eliminated using rotation, identifying the conic becomes more straightforward:

• Circle: The equation will have equal coefficients for \(x^2\) and \(y^2\).
• Ellipse: Coefficients of \(x^2\) and \(y^2\) are positive but not equal.
• Parabola: Only one of \(x^2\) or \(y^2\) appears, with possibly other terms.
• Hyperbola: Coefficients of \(x^2\) and \(y^2\) are opposite in signs.

Identifying these sections helps in visualizing and solving problems involving intersections and systems of equations.

Angle of Rotation Calculation

When you perform rotation to eliminate the \(xy\) term from a conic equation, it’s important to calculate the precise angle of rotation. Generally, this angle allows you to realign the coordinate system effectively.The key formula is \(\tan(2\theta) = \frac{B}{A-C}\), where \( A\), \(B\), and \(C\) are coefficients from the original conic equation. This equation arises from the principle of aligning the transformed axes with the principal axes of the conic shape. In some cases, the equation \(\tan(2\theta)\) might give an undefined result. When this occurs, it means the angle \(\theta\) is either \(\frac{\pi}{4}\) (45 degrees) or \(-\frac{\pi}{4}\), which are common angles for diagonal transformations on the plane to reduce or eliminate the \(xy\) term. After finding \(\theta\), substitute it back into the rotation formulas to transform \(x\) and \(y\), helping ensure that the resulting new coordinates simplify the equation effectively.