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

Step-by-Step Solution

Answer

The graph is a parabola. Use rotation of axes to set the standard parabola form and sketch it.

Step 1: Expand and rearrange the equation

Start by expanding the right-hand side of the equation. The term \( 100(x-y-1) \) expands to \( 100x - 100y - 100 \). Rewrite the original equation as: \( 9x^2 - 24xy + 16y^2 = 100x - 100y - 100 \). Rearrange all terms to one side to get \( 9x^2 - 24xy + 16y^2 - 100x + 100y + 100 = 0 \).

Step 2: Determine the discriminant

For a general conic section equation \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \), the discriminant \( \Delta \) is given by \( \Delta = B^2 - 4AC \). For our equation, \( A = 9 \), \( B = -24 \), and \( C = 16 \). Calculate \( \Delta = (-24)^2 - 4 \times 9 \times 16 = 576 - 576 = 0 \). Since \( \Delta = 0 \), the graph is a parabola.

Step 3: Use rotation of axes to eliminate the \(xy\)-term

Use the rotation formula \( \tan(2\theta) = \frac{B}{A-C} \) to find the angle \( \theta \). Substituting \( B = -24 \), \( A = 9 \), and \( C = 16 \), we get \( \tan(2\theta) = \frac{-24}{9-16} = \frac{-24}{-7} = \frac{24}{7} \). Calculate \( \theta = \frac{1}{2} \tan^{-1}\left(\frac{24}{7}\right) \). This \( \theta \) will give us the rotation needed to eliminate the \( xy \)-term.

Step 4: Substitute the rotation formulas

The rotation formulas for \( x' \) and \( y' \) are \( x = x' \cos\theta - y'\sin\theta \) and \( y = x'\sin\theta + y'\cos\theta \). Substitute these into the original equation to eliminate the \( xy \)-term, leading to a new quadratic in terms of \( x' \) and \( y' \). Perform this substitution to find the new form of the equation.

Step 5: Sketch the parabola

After rotation and simplification, the equation in terms of \( x' \) and \( y' \) will represent a standard form of a parabola. Sketch the parabola based on the transformed equation. Note any shifts and the direction of the axis of the parabola, using the values calculated for the rotation.

Key Concepts

Parabola

In the study of conic sections, a **parabola** is a curve where any point is at an equal distance from a fixed point known as the "focus" and a line called the "directrix". It's important to recognize parabolas in equations, and for quadratic equations, the format generally takes the form:

• \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \)

In this exercise, after simplifying and rearranging, the discriminant helps us identify that the given equation indeed represents a parabola. Here the discriminant is zero (i.e., \( \Delta = 0 \)), which means that the graph of the equation is a parabola. Recognizing the form and using algebraic techniques to identify these properties is key when solving such problems.

Discriminant

The **discriminant** plays a crucial role in understanding the nature of conic sections in quadratic equations. It is given by the formula:

• \( \Delta = B^2 - 4AC \)

Here, \( A \), \( B \), and \( C \) are coefficients from the equation \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). The discriminant helps to determine the type of conic section:

• \( \Delta > 0 \) implies a hyperbola.
• \( \Delta = 0 \) implies a parabola.
• \( \Delta < 0 \) implies an ellipse.

In the given equation, \( \Delta = 0 \), confirming it as a parabola. Calculating the discriminant provides immediate insights into which conic section one is dealing with, making it an efficient first step in many problems.

Rotation of Axes

The **rotation of axes** is a very handy technique for simplifying complex conic equations by eliminating the \(xy\)-term. This is particularly beneficial as it allows us to represent the conic in its standard form. In this exercise, we use the formula:

• \( \tan(2\theta) = \frac{B}{A-C} \)

Solving for \( \theta \) gives the angle, altering the axes to eliminate the \( xy \)-term. Substituting \( B = -24 \), \( A = 9 \), and \( C = 16 \), results in \( \tan(2\theta) = \frac{24}{7} \). Calculating \( \theta \), we then use the rotation formulas:

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

Inserting these into the original equation repositions the axes such that the \(xy\)-term vanishes, simplifying further analysis of the conic's properties.

Elimination of xy-term

To achieve a more manageable form of a conic section equation, we often aim to eliminate the **xy-term**. The previously discussed rotation of axes is employed precisely for this purpose. By finding an appropriate angle \(\theta\), the substitution of the coordinates allows the transformation:

• from \(x, y\) to new axes \(x', y'\)

This substitution is crucial as it can often transform an unwieldy quadratic equation into a standard form that is far easier to work with. In effect, it spells out the parabola's innate orientation and central position without the interference of cross terms.
Following this process, the equation is reduced to a more traditional form where analysis and graph sketching become more straightforward. This step ensures that the properties of the conic section are clearly decipherable, enhancing both understanding and visual representation.