Problem 26 - Algebra and Trigonometry

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. $$(7 x+24 y)^{2}=600 x-175 y+25$$

Step-by-Step Solution

Answer

The graph is a degenerate parabola, represented by two parallel lines.

Step 1: Expand and Rearrange Equation

Begin by expanding the left side of the given equation \((7x + 24y)^2 = 600x - 175y + 25\). Expanding \((7x + 24y)^2\) gives us \(49x^2 + 2\cdot7\cdot24\cdot xy + 576y^2 = 49x^2 + 336xy + 576y^2\). Rearrange the equation as \(49x^2 + 336xy + 576y^2 = 600x - 175y + 25\).

Step 2: Match with Conic Equation Form

The standard second-degree equation form is \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). For our equation, \(A = 49\), \(B = 336\), \(C = 576\), \(D = -600\), \(E = 175\), \(F = -25\).

Step 3: Compute the Discriminant

The discriminant for conics is given by \(B^2 - 4AC\). Substitute \(A = 49\), \(B = 336\), and \(C = 576\) into the formula: \(B^2 - 4AC = 336^2 - 4 \times 49 \times 576\). Calculate \(336^2 = 112896\) and \(4 \times 49 \times 576 = 112896\). Therefore, the discriminant is \(0\).

Step 4: Identify the Conic

Since the discriminant \(B^2 - 4AC = 0\), the equation represents a parabola. This indicates the equation is a degenerate form of a quadratic conic.

Step 5: Apply Axis Rotation

To eliminate the \(xy\)-term, use the formula \(tan(2\theta) = \frac{B}{A-C}\) for the rotation angle \(\theta\). Since \(B = 336\) and \(A = 49\), \(C = 576\), we get \(tan(2\theta) = \frac{336}{49-576} = \frac{336}{-527}\). Calculate \(2\theta\) and find \(\theta\). Simplify to get new axes coordinates.

Step 6: Graph the Parabola

After rotating the axes, translate the adjusted equation. Since the original equation is degenerate and resolves to straight lines \(7x + 24y = \pm\sqrt{25} = \pm5\), graph these lines. These lines are parallel, indicating a degenerate conic.

Key Concepts

Discriminant

The discriminant plays a crucial role in determining the type of conic section represented by a second-degree equation. It is given by the formula \(B^2 - 4AC\), where \(A\), \(B\), and \(C\) are coefficients from the standard second-degree equation form: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\).

• If the discriminant is greater than zero, the conic is a hyperbola.
• If it equals zero, as in our exercise, the conic is a parabola.
• If it is less than zero, the conic is an ellipse.

In our exercise, after calculating \(B^2 - 4AC\) with \(A = 49\), \(B = 336\), and \(C = 576\), we found it equals zero. Therefore, the equation corresponds to a degenerate parabola. This concept helps simplify complex equations and provides insights into their graphical nature.

Axis Rotation

Axis rotation is a technique used to eliminate the \(xy\)-term in a second-degree equation of a conic section. This process simplifies the equation, making it easier to identify and graph the conic section.

The rotation angle \(\theta\) is found using the formula: \(\tan(2\theta) = \frac{B}{A-C}\). In our example, \(B = 336\), \(A = 49\), and \(C = 576\), so \(\tan(2\theta) = \frac{336}{-527}\).
Calculating \(\theta\) helps in determining new coordinates aligned with the axes. This change often reveals the true nature of the conic section when it might be obscured in its initial form.

Axis rotation can also involve trigonometric transformations to translate coordinates into a new reference frame, effectively removing the \(xy\)-term and refining the depiction of the conic section.

Parabola

A parabola is a distinctive type of conic section characterized by its "U"-shaped curve. However, in the realm of conic sections, a parabola's defining feature is that every point is equidistant from a fixed point, called the focus, and a fixed line, called the directrix. When the discriminant \(B^2 - 4AC\) equals zero, the given equation represents either a standard or a degenerate parabola.
In our exercise, the equation resolved to straight lines, indicating a degenerate conic – where what we think of as a parabola becomes two intersecting lines instead of a curved "U" shape.

• This can occur when an expected curve "collapses" due to the specific coefficients.
• Hence, interpreting the structure as straight lines instead of curves becomes necessary.

Degenerate Conic

Degenerate conics are a unique case where the conic section does not appear in its typical shape. Instead, it may manifest as a point, a line, or intersecting lines. This occurs when certain conditions are met, such as the discriminant equating to zero in a supposedly parabolic equation.
In our case, the equation evolved into two straight lines: \(7x + 24y = \pm5\). These parallel lines illustrate how degenerate conics can represent unusual occurrences in graphing conic sections.

• This situation often requires a deeper understanding of the interaction of coefficients and the geometric interpretation of the conic equation.
• Degenerate conics can provide insight into the constraints and limitations present within the mathematical model.