Problem 25 - 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. $$52 x^{2}+72 x y+73 y^{2}=40 x-30 y+75$$

Step-by-Step Solution

Answer

The equation is an ellipse after rotation, having no xy-term.

Step 1: Identifying Discriminant for Conic Section

The general form of a conic section is given by the equation \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). For our equation, these coefficients are: \(A = 52\), \(B = 72\), \(C = 73\), \(D = -40\), \(E = 30\), \(F = -75\). The discriminant \(\Delta = B^2 - 4AC\) helps determine the type of conic. Calculate \(\Delta = 72^2 - 4(52)(73)\).

Step 2: Calculating Discriminant

Compute the values: \(72^2 = 5184\) and \(4 \times 52 \times 73 = 15184\). Thus, \(\Delta = 5184 - 15184 = -10000\).

Step 3: Type of Conic Section

Since \(\Delta = -10000 < 0\), the given equation represents an ellipse.

Step 4: Performing Rotation of Axes

To eliminate the \(xy\)-term, use the formula for an angle \(\theta\) where \(\tan(2\theta) = \frac{B}{A-C}\). Thus, \(\tan(2\theta) = \frac{72}{52-73} = -\frac{72}{21}\). To find \(\theta\), use \(\theta = \frac{1}{2} \tan^{-1}\left(-\frac{72}{21}\right)\).

Step 5: Calculating Rotation Angle

Find \(\theta\) using a calculator: \(\theta = \frac{1}{2} \tan^{-1}\left(-\frac{72}{21}\right)\). Approximating, \(\tan^{-1}\left(-\frac{72}{21}\right) \approx -73.74°\), so \(\theta \approx -36.87°\).

Step 6: Sketching the Graph

Given that the equation is an ellipse, it will look like an elongated circle. After eliminating the \(xy\)-term using the rotation, sketch this roughly as a standard ellipse, noting its orientation from the calculated \(\theta\).

Key Concepts

Ellipse Identification

Determining whether a conic equation represents an ellipse is essential for understanding its characteristics. To do so, we use the discriminant, which is given by the formula

• \( \Delta = B^2 - 4AC \).

For the equation provided, where \( A = 52, B = 72, \) and \( C = 73 \), substituting these values into the equation gives \( \Delta = 72^2 - 4(52)(73) = -10000 \). If \( \Delta < 0 \), the conic is an ellipse.
Identifying the type of conic this way helps us understand its geometric nature. An ellipse represents a set of points such that the sum of the distances from any point on the ellipse to two fixed points (foci) is constant. Knowing this, you can focus on other properties, like its axes, to sketch and analyze the shape efficiently. Remember, ellipses appear flat or elongated, depending on the coefficients, and the negative value for \( \Delta \) confirms this in our problem.

Conic Section Rotation

Rotating a conic section's axes can eliminate any cross term \(xy\) in its equation, simplifying the form to make calculations easier. The angle \( \theta \) necessary for rotation is found using the formula:

• \( \tan(2\theta) = \frac{B}{A-C} \).

For our problem, \( \tan(2\theta) = \frac{72}{52-73} = -\frac{72}{21} \).
Using this, you calculate \( \theta = \frac{1}{2} \tan^{-1}\left(-\frac{72}{21}\right) \). This approximates to \( \theta \approx -36.87^\circ \).
Performing this rotation clarifies the conic's orientation and symmetries better, especially when you sketch it. This angle indicates how much to "tilt" the coordinate system so the ellipse aligns with one of the axes. By doing so, any effects from the original \(xy\) term are neutralized, making visualizing and further calculating with the conic manageable.

Graphical Representation of Conics

Sketching the graph of a conic section, such as an ellipse, provides a visual understanding of its properties and orientation. After identifying the conic as an ellipse and rotating the axes, plotting requires some attention to the conic's overall shape and direction.
Remember:

• An ellipse looks like a squished circle, elongated along its major axis.
• In this case, the negative discriminant confirms a standard ellipse shape.
• With the angle \(\theta\) calculated, align the major axis according to this rotation.

Plotting the ellipse helps grasp how it interacts within a coordinate system, especially after making it aligned post-rotation. It translates theoretical understanding into a visual form, helping you predict how changes in coefficients affect the conic further. Graphically, you should ensure the major and minor axes are correctly represented relative to their calculated orientations for accuracy.