Problem 15 - Precalculus: Mathematical for Calculus

Question

(a)Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$x^{2}+2 x y+y^{2}+x-y=0$$

Step-by-Step Solution

Answer

The graph is a parabola, and rotating the axes eliminates the \( xy \)-term, revealing its orientation.

Step 1: Identify the Conic Section

Given the general equation of a conic section: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]For the equation \( x^2 + 2xy + y^2 + x - y = 0 \), we identify \( A = 1 \), \( B = 2 \), and \( C = 1 \). To determine the type of conic section, calculate the discriminant: \[ \Delta = B^2 - 4AC \]Substitute the values:\[ \Delta = 2^2 - 4(1)(1) = 4 - 4 = 0 \]Since \( \Delta = 0 \), the conic is a parabola.

Step 2: Determine the Rotation Angle

When \( B eq 0 \) in the form \( Bxy \), we can rotate the axes to eliminate the \( xy \)-term. The angle \( \theta \) for the rotation is given by:\[ \cot(2\theta) = \frac{A-C}{B} \]Here, \( A = 1 \), \( C = 1 \), and \( B = 2 \). Hence:\[ \cot(2\theta) = \frac{1-1}{2} = 0 \]\( \cot(2\theta) = 0 \) implies \( 2\theta = \frac{\pi}{2} \) or \( \theta = \frac{\pi}{4} \). Rotate the axes by \( 45^\circ \) to eliminate the \( xy \)-term.

Step 3: Rotate the Axes

Using the angle \( \theta = \frac{\pi}{4} \), perform the rotation with the transformations:\[ x = x'\cos(\theta) - y'\sin(\theta) \]\[ y = x'\sin(\theta) + y'\cos(\theta) \]With \( \theta = \frac{\pi}{4} \), calculate:\[ x = \frac{1}{\sqrt{2}}(x' - y') \]\[ y = \frac{1}{\sqrt{2}}(x' + y') \]Substitute these in the original equation to obtain an equation in terms of \( x' \) and \( y' \) without the \( x'y' \)-term.

Step 4: Simplify the Transformed Equation

Substitute and simplify:- Substitute in the equation: \( (\frac{1}{\sqrt{2}}(x' - y'))^2 + 2(\frac{1}{\sqrt{2}}(x' - y'))(\frac{1}{\sqrt{2}}(x' + y')) + (\frac{1}{\sqrt{2}}(x' + y'))^2 + (\frac{1}{\sqrt{2}}(x' - y')) - (\frac{1}{\sqrt{2}}(x' + y')) = 0 \).- Further simplify to eliminate terms and achieve the new basic equation form in terms of \( x' \) and \( y' \).- By simplification, you should get a simpler expression, representing the rotated parabola.

Step 5: Sketch the Graph

The resulting equation simplifies to a form that’s easy for plotting. Recognize the new patterns, such as opening direction and vertex position.By rotating the graph \( 45^\circ \) counterclockwise and solving/simplifying the equation, sketch this parabola with the corresponding vertices and orientations.

Key Concepts

Discriminant

In the context of conic sections, the discriminant provides significant insight into the nature of the curve represented by a given quadratic equation. For an equation in the form \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \), the discriminant is calculated as \( \Delta = B^2 - 4AC \). This value helps determine whether the conic section is a parabola, ellipse, or hyperbola:

• If \( \Delta < 0 \), the conic is an ellipse.
• If \( \Delta = 0 \), we have a parabola.
• If \( \Delta > 0 \), the form is a hyperbola.

In our problem, after substituting \( A = 1 \), \( B = 2 \), and \( C = 1 \) into the discriminant formula, we found \( \Delta = 2^2 - 4(1)(1) = 0 \). Therefore, the equation is identified as a parabola.

Rotation of Axes

The rotation of axes is a valuable technique for simplifying the equation of a conic section, especially when the equation includes a mixed term, such as \( xy \). To eliminate this term and simplify the equation, we use the rotation formulas. If a term \( Bxy \) exists, there is a suitable angle \( \theta \) that can transform the axis in such a way that the mixed term disappears.The angle \( \theta \) for rotation is calculated using \( \cot(2\theta) = \frac{A-C}{B} \). For our problem, this calculation with \( A = 1 \), \( C = 1 \), and \( B = 2 \) resulted in \( \cot(2\theta) = 0 \), leading to \( 2\theta = \frac{\pi}{2} \) or \( \theta = \frac{\pi}{4} \). Such a rotation converts the axes and can simplify the analysis of the conic.

Parabola

A parabola is a unique type of conic section characterized by its U-shape and symmetry. Parabolas can open upwards, downwards, left, or right. The standard form of a parabola can be written depending on its orientation. One feature of parabolas is that they can be easily recognized by the equation properties:

• When the discriminant \( \Delta = 0 \), it confirms a parabolic structure.
• The equation shows a single squared variable without the other squared or as the coefficient, representing the U-shape.

In the rotated coordinate system, we simplify the parabola's equation to better visualize and identify features like vertices and orientation. This process makes sketching and further analysis much easier.

Graphing

Graphing conic sections involves understanding the equations' form and applying transformations to achieve an explicit visualization. For graphing a parabola, particularly after axis rotation, one should consider the new simplified equation. This aids in pinpointing the parabolic curve's characteristics, such as its direction, vertex, and axis of symmetry. To graph effectively, follow these steps:

• Identify key elements: vertex, axis of symmetry, and focus (if needed).
• Use intercepts and key points to plot accurately.
• For rotated axes, adjust for the new orientation by reflecting changes in the sketch.

By attending to these details, one can produce a clear and accurate representation of the parabola from the transformed coordinate system.