Problem 27 - Algebra and Trigonometry

Question

\(15-28=(a)\) 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. $$ 52 x^{2}+72 x y+73 y^{2}=40 x-30 y+75 $$

Step-by-Step Solution

Answer

(a) The graph is an ellipse. (b) Use a rotation to eliminate \(xy\). (c) Sketch the ellipse.

Step 1: Transforming the Equation

First, rearrange the equation to standard general form: \[52x^2 + 72xy + 73y^2 - 40x + 30y - 75 = 0.\] Identify the coefficients: \(A = 52\), \(B = 72\), \(C = 73\).

Step 2: Evaluate the Discriminant

To determine the type of conic section, use the discriminant \(D = B^2 - 4AC\). Substitute the values:\[D = 72^2 - 4 \times 52 \times 73.\]Calculate:\[D = 5184 - 15184 = -10000.\] Since \(D < 0\), the graph represents an ellipse.

Step 3: Determine Rotation Angle

For axes rotation, use the formula \(\tan(2\theta) = \frac{B}{A - C}\). Substituting values:\[\tan(2\theta) = \frac{72}{52 - 73} = \frac{72}{-21}.\]Calculate:\[2\theta = \tan^{-1}\left(-\frac{72}{21}\right).\] Solve to find \(\theta\).

Step 4: Apply Rotation of Axes

Using \(\theta\) from the previous step, the new coordinates \((x', y')\) are selected by:\[x = x' \cos\theta - y' \sin\theta,\]\[y = x' \sin\theta + y' \cos\theta.\]Substitute these into the original equation to eliminate the \(xy\)-term.

Step 5: Sketch the Graph

Graph the resulting equation from step 4, which should reduce to a standard ellipse form. That is, you will draw the ellipse based on the new axes and transformed equation while ensuring it aligns correctly according to rotation.

Key Concepts

Discriminant Method

The discriminant method is a powerful tool for identifying the type of conic section represented by a given equation. To determine if a graph is a parabola, ellipse, or hyperbola, you need to evaluate the discriminant, denoted by the formula: \[D = B^2 - 4AC\]where \(A\), \(B\), and \(C\) are the coefficients of \(x^2\), \(xy\), and \(y^2\) respectively in the equation of the conic.
Based on the value of \(D\):

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

Evaluating \(D\) is straightforward and provides quick insights into the nature of the conic section, allowing us to classify the graph accurately.

Rotation of Axes

Sometimes, when dealing with conic sections, the presence of an \(xy\)-term can make the graphing and classification process more complicated. To simplify this, we use the technique of rotation of axes. This mathematical procedure involves rotating the coordinate axes to eliminate the \(xy\)-term in the conic equation.
To find the angle of rotation \(\theta\), we use the relation:\[\tan(2\theta) = \frac{B}{A - C}\]This gives us the angle at which we need to rotate our axes to remove the \(xy\)-term. Once \(\theta\) is determined, substitute \(x\) and \(y\) using the transformations:

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

This changes the original equation to one that is easier to manage, especially for graphing purposes.

Ellipse Graphing

Graphing an ellipse requires understanding its standard form. Once the conic equation is transformed and simplified, the resulting form should resemble one of the general standard forms. For ellipses, the standard forms are:

• \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\)
• \(\frac{(y-k)^2}{a^2} + \frac{(x-h)^2}{b^2} = 1\)

where \((h, k)\) is the center of the ellipse, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis.
To sketch the graph of an ellipse:

• Identify the center \((h, k)\).
• Draw the principal axes (major and minor).
• Plot the vertices and co-vertices based on the axes lengths.
• Connect these points smoothly to form an oval shape.

Understanding how to graph an ellipse gives visual insight into its orientation and size.

Parabola vs Ellipse vs Hyperbola

When dealing with conic sections, it's essential to distinguish between a parabola, an ellipse, and a hyperbola as they have distinct shapes and properties. Here's how they compare:
- **Parabola**: Typically U-shaped, a parabola opens either upwards, downwards, or sideways. It's defined by its vertex, axis of symmetry, and a directrix. Parabolas have one focus. - **Ellipse**: An ellipse looks like an elongated circle. It's defined by two foci and an oval shape where all points on the curve maintain the sum of distances to the foci as constant. Ellipses are closed curves. - **Hyperbola**: Hyperbolas have two separate curves opening in opposite directions. They are defined by two foci and an axis of symmetry. The difference in distances from any point on the hyperbola to the foci is constant. Differences lay in their geometric properties and where they appear in physical phenomena, such as parabolic paths of projectiles, elliptical orbits of planets, and hyperbolic paths in certain types of navigation systems.