Problem 26 - 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. $$9 x^{2}-24 x y+16 y^{2}=100(x-y-1)$$

Step-by-Step Solution

Answer

The graph is a parabola; use rotation to eliminate the xy-term and sketch it on rotated axes.

Step 1: Simplify the Equation

First, expand and simplify the equation \(9x^2 - 24xy + 16y^2 = 100(x-y-1)\). This gives \[9x^2 - 24xy + 16y^2 = 100x - 100y - 100\]. Rearrange the terms to obtain \[9x^2 - 24xy + 16y^2 - 100x + 100y + 100 = 0\].

Step 2: Identify the Conic Type Using the Discriminant

The discriminant \(D\) for a conic section given in the general form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\) is \(D = B^2 - 4AC\). Here, \(A = 9\), \(B = -24\), and \(C = 16\). Calculate the discriminant: \[D = (-24)^2 - 4(9)(16) = 576 - 576 = 0\]. A zero discriminant indicates the conic is a parabola.

Step 3: Eliminate the xy-term by Rotation of Axes

Using the formula \( \tan 2\theta = \frac{B}{A-C} \), where \(B = -24\), \(A = 9\), and \(C = 16\), calculate \( \tan 2\theta = \frac{-24}{9-16} = \frac{-24}{-7} = \frac{24}{7} \). Solve \(2\theta\) to find \(\theta \). Use the identity \( \tan 2\theta = \frac{24}{7} \) to find \(\theta\), leading to a new axis orientation.

Step 4: Sketch the Graph

After the rotation of axes, write the equation in its new form which will no longer have an \(xy\) term and will exactly describe a parabola. Sketch the graph using this form on the rotated axes oriented at angle \(\theta\).

Key Concepts

Discriminant

The discriminant is a crucial tool in determining the type of conic section represented by a quadratic equation. In the general quadratic form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the discriminant is calculated using the formula \(D = B^2 - 4AC\). The value of \(D\) helps to distinguish between different conic sections:

• If \(D > 0\), the conic is a hyperbola.
• If \(D = 0\), it is a parabola.
• If \(D < 0\), the conic section is an ellipse.

In the current example, we have \(A = 9\), \(B = -24\), and \(C = 16\). Calculating the discriminant gives \(D = (-24)^2 - 4\times 9 \times 16 = 576 - 576 = 0\). Therefore, the conic section described by the equation is a parabola. Understanding the discriminant can quickly tell us about the overall shape without fully solving for the conic.

Rotation of Axes

When a conic section in an equation contains an \(xy\)-term, this indicates that the axes are not aligned with the standard coordinate axes. To simplify such equations and eliminate the \(xy\)-term, we can use a rotation of axes. This involves rotating the coordinate system to a new set of axes that realigns the equation to a simpler form.To determine the angle \(\theta\) for this rotation, use the formula \(\tan 2\theta = \frac{B}{A-C}\). In this case:

• \(B = -24\)
• \(A = 9\)
• \(C = 16\)

Substituting these values gives \(\tan 2\theta = \frac{-24}{9-16} = \frac{24}{7}\). Solving for \(\theta\) from the tangent function allows us to rotate the coordinates so that the new axes eliminate the \(xy\)-term. This simplifies graphing and allows a clear visual representation of the parabola.

Parabola

A parabola is one of the most familiar conic sections. It is defined as the set of all points in a plane equidistant from a fixed point called the focus and a line called the directrix. Parabolas have a characteristic U-shaped curve that can open either upward, downward, or sideways, depending on the orientation of the axis.In the simplified form that does not contain an \(xy\)-term, the equation of a parabola can be easily identified and transformed to fit standard forms like \((x-h)^2 = 4p(y-k)\) or \((y-k)^2 = 4p(x-h)\) depending on its opening direction. The vertex form is particularly useful for sketching with the vertex at \((h,k)\) and the focus affecting the width and direction of the opening.

Graph Sketching

After simplifying the equation and identifying the conic as a parabola, the next step is to graph it. Graph sketching begins with determining key features such as the vertex, axis of symmetry, direction of opening, and points of intersection if any.Following the rotation of axes:

• Use the new equation form without \(xy\)-term for sketching.
• Find the vertex, which often lies at the focus or at the point derived from transformation equations.
• Determining the direction and scale of the parabola helps visualize its full span across axes.
• Plot additional points if needed to confirm the shape.

This visualization is crucial for fully understanding the geometric implications of the parabola in question. Equations in their simplest form make it easier to relate to real-world scenarios and applications of parabolic structures.