Problem 20 - 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. $$25 x^{2}-120 x y+144 y^{2}-156 x-65 y=0$$

Step-by-Step Solution

Answer

The equation's graph is a parabola. Transformation eliminates the \(xy\)-term, making it easier to sketch.

Step 1: Determine the Conic Type Using the Discriminant

To determine the type of conic, we use the discriminant formula for conics, which is given by \( B^2 - 4AC \). Here, the given equation is \( 25x^{2} - 120xy + 144y^{2} - 156x - 65y = 0 \). Identify the coefficients: \( A = 25 \), \( B = -120 \), and \( C = 144 \). Compute the discriminant: \( B^2 - 4AC = (-120)^2 - 4(25)(144) \). Calculate this step by step: \( B^2 = 14400 \) and \( 4AC = 14400 \), so \( B^2 - 4AC = 0 \). Since the discriminant is zero, the conic is a parabola.

Step 2: Apply Rotation of Axes to Eliminate the \(xy\)-term

To eliminate the \(xy\)-term, we must rotate the axes by an angle \( \theta \) such that \( \tan(2\theta) = \frac{B}{A-C} \). Here, \( B = -120 \), \( A = 25 \), and \( C = 144 \), so \( \tan(2\theta) = \frac{-120}{25-144} = \frac{-120}{-119} \approx 1.0084 \). Solving \( \tan(2\theta) = 1.0084 \) gives \( \theta \approx 22.62^\circ \). Use this \( \theta \) to transform the equation to one with no \(xy\)-term using rotation formulas: \( x = x' \cos\theta - y' \sin\theta \) and \( y = x' \sin\theta + y' \cos\theta \). Substitute these in the original equation and simplify.

Step 3: Sketch the Graph of the Transformed Equation

After eliminating the \(xy\)-term through rotation, you simplify the equation into a standard form that reveals the nature of the graph. Due to the transformation, the parabola will be oriented along one of the axes but with no \(xy\)-term. Sketch this transformed standard form, identifying key features like vertex, axis of symmetry, and direction of opening based on simplified terms and coefficients. Though computations can be extensive, the general shape after rotation will be sketched based on approximate values and the knowledge that it is a parabola.

Key Concepts

Discriminant in Conics

When dealing with conic sections, one of the most important tools available to us is the discriminant. This allows us to quickly determine the type of conic we are examining. The discriminant for conic sections is given by the formula \( B^2 - 4AC \). Here, \( A \), \( B \), and \( C \) are coefficients from the general quadratic equation of a conic: \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). By evaluating \( B^2 - 4AC \), you can ascertain the conic's nature very efficiently:

• If \( B^2 - 4AC < 0 \), the graph is an ellipse.
• If \( B^2 - 4AC = 0 \), the graph is a parabola.
• If \( B^2 - 4AC > 0 \), the graph is a hyperbola.

For the problem at hand, with coefficients \( A = 25 \), \( B = -120 \), and \( C = 144 \), calculating the discriminant yields \( 0 \). This conclusively identifies the conic as being a parabola, simplifying the classification process immensely.

Rotation of Axes

In conics, the presence of an \( xy \)-term complicates the equation, as it suggests a rotation in the graph. To simplify it, we rotate the coordinate axes to eliminate this term. The essential step involves finding an angle \( \theta \) through which we rotate the axes. The angle can be calculated using the formula \( \tan(2\theta) = \frac{B}{A-C} \), where \( A \), \( B \), and \( C \) are the coefficients of the original quadratic equation.

For the given equation \( 25x^2 - 120xy + 144y^2 - 156x - 65y = 0 \), the values \( A = 25 \), \( B = -120 \), and \( C = 144 \) produce \( \tan(2\theta) = \frac{-120}{25-144} = \frac{-120}{-119} \). Calculating this gives \( \theta \approx 22.62^\circ \).

After finding \( \theta \), replace \( x \) and \( y \) in the original equation with the rotated variables as follows:

• \( x = x' \cos\theta - y' \sin\theta \)
• \( y = x' \sin\theta + y' \cos\theta \)

This transformation removes the \( xy \)-term, simplifying the work needed to analyze and sketch the graph.

Graphing Parabolas

Once the \( xy \)-term is eliminated through rotation, we will have a simpler equation, often making graphing tasks more approachable. A rotated parabola translates through the rotated axes, preserving its essential properties like the vertex, axis of symmetry, and direction of opening. The equation might not look like the standard `\( y = ax^2 + bx + c \)` form after rotation, yet the structure will tell you much about the parabola's orientation.

In our case, knowing the conic is a parabola, you will sketch it respecting these properties:

• The vertex is the peak or lowest point of the curve, easily identified in the transformed equation.
• The axis of symmetry will run through the vertex, guiding the parabola's opening direction.
• Typically, parabolas open either upward, downward, left, or right, depending on the coefficients in their transformed equation.

Based on knowledge of the transformed equation's form, draw the graph confirming its parabola nature after rotation, making sure it aligns properly in the newly configured frame.