Problem 23 - Precalculus Mathematics 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 \(xy\)-term. (c) Sketch the graph. $$x^{2}+2 x y+y^{2}+x-y=0$$

Step-by-Step Solution

Answer

The equation represents a parabola.

Step 1: Introduction to the Equation

We are given the equation \(x^2 + 2xy + y^2 + x - y = 0\). Our goal is to determine whether the graph is a parabola, an ellipse, or a hyperbola, use rotation to eliminate the \(xy\)-term, and finally sketch the graph.

Step 1: Determine the Conic Section Using the Discriminant

The discriminant of a general second-degree equation \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\) is given by \(B^2 - 4AC\). Here, \(A = 1\), \(B = 2\), and \(C = 1\). Thus, the discriminant is \(2^2 - 4 \times 1 \times 1 = 4 - 4 = 0\). Since the discriminant is 0, the graph represents a parabola.

Step 2: Use Rotation of Axes to Eliminate the xy-Term

Since the \(xy\)-term is present, we use axis rotation to eliminate it. The angle \(\theta\) for rotation is given by \(\tan(2\theta) = \frac{B}{A-C}\), so \(\tan(2\theta) = \frac{2}{1-1} = \text{undefined}\). This means \(2\theta = 90^\circ\), so \(\theta = 45^\circ\). We substitute \(x = x'\cos(45^\circ) - y'\sin(45^\circ)\) and \(y = x'\sin(45^\circ) + y'\cos(45^\circ)\) into the equation, simplifying using \(\cos(45^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2}\).

Step 3: Substitute and Simplify the Equation

After substituting \(x = \frac{\sqrt{2}}{2}(x' - y')\) and \(y = \frac{\sqrt{2}}{2}(x' + y')\) into the original equation, simplify to eliminate the \(xy\)-term. After simplification, the equation appears in a new form without the \(xy'\)-term. Solve and re-arrange terms to get the final conic in terms of \(x'\) and \(y'\).

Step 4: Sketch the Graph

Based on the simplified equation in terms of \(x'\) and \(y'\), sketch the graph. As we identified, it is a parabola. The simplified equation provides the orientation and characteristic of this parabola. Plot the vertex and orientation after the transformation.

Key Concepts

Discriminant in Conics

Understanding conic sections is crucial for determining the shape of a graph. The discriminant is a mathematical tool that helps us do this. In any general second-degree equation of the form \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \), the discriminant is given by \( B^2 - 4AC \). This value determines the type of conic section you are dealing with:

• If the discriminant is less than 0, the conic is an ellipse.
• If the discriminant equals 0, it is a parabola.
• If it is greater than 0, the conic is a hyperbola.

For the equation given \( x^2 + 2xy + y^2 + x - y = 0 \), the discriminant calculation \( 2^2 - 4 \times 1 \times 1 = 0 \) confirms it is a parabola. This tells us that the graph will have a single directrix and focus aligned in a way typical for parabolas.

Rotation of Axes

Converting a conic section equation into a simpler form can often involve rotating the axes. This eliminates the \( xy \)-term, making the equation easier to understand and work with. The angle of rotation \( \theta \) is determined using \( \tan(2\theta) = \frac{B}{A-C} \). When substituting our specific values to eliminate the \( xy \)-term, we get \( \tan(2\theta) = \frac{2}{1-1} = \) undefined. This suggests the angle \( 2\theta = 90^\circ \), resulting in \( \theta = 45^\circ \).

Subsequently, we use the following transformations for rotation:

• \( x = x' \cos(45^\circ) - y' \sin(45^\circ) \)
• \( y = x' \sin(45^\circ) + y' \cos(45^\circ) \)

Utilizing \( \cos(45^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2} \), you can simplify the equation by substituting \( x \) and \( y \) with the transformed coordinates. This results in a new, simpler equation, without the \( xy \)-term.

Parabola Graphing

Graphing a parabola involves understanding its key features, which typically include a vertex, axis of symmetry, and a directrix. After using rotation of the axes to remove the \(xy\)-term in an equation, the goal is to graph the resulting simplified parabola. Consider the standard form of a parabola:\(y=ax^2+bx+c\), where the graph reveals the direction in which it opens - depending on the coefficients.

For the transformed equation we've derived, plot the essential characteristics:

• Identify the vertex, which acts as the turning point.
• Determine the axis of symmetry, a vertical line through the vertex that divides the parabola into two mirror-image halves.
• Decide the direction the parabola opens based on its orientation.

With these steps, you gain a clearer picture of the parabola's geometry despite the initial complexity of the equation. This deepens comprehension and aids in visually representing mathematical concepts effectively.