Problem 19 - 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. $$11 x^{2}-24 x y+4 y^{2}+20=0$$

Step-by-Step Solution

Answer

The graph is a hyperbola with a significant rotation to eliminate the \(xy\)-term.

Step 1: Identify the Coefficients

First, write the given equation in standard form: \(11x^2 - 24xy + 4y^2 + 20 = 0\). Identify the coefficients: \(A = 11\), \(B = -24\), and \(C = 4\).

Step 2: Calculate the Discriminant

The discriminant for a conic section is given by the formula \(D = B^2 - 4AC\). Substitute the values into the formula: \(D = (-24)^2 - 4 \cdot 11 \cdot 4\). Calculate \(D\): \(D = 576 - 176 = 400\).

Step 3: Classify the Conic Using the Discriminant

The nature of the conic section is determined by the discriminant \(D\). If \(D < 0\), it is an ellipse; if \(D = 0\), it is a parabola; and if \(D > 0\), it is a hyperbola. Since \(D = 400 > 0\), the graph is a hyperbola.

Step 4: Determine the Rotation Angle

Use the formula \(\cot 2\theta = \frac{A-C}{B}\) to find the angle \(\theta\). Substitute the values \(A = 11\), \(C = 4\), and \(B = -24\): \(\cot 2\theta = \frac{11-4}{-24} = \frac{7}{-24}\). Find \(2\theta\) using \(\tan 2\theta = -\frac{24}{7}\).

Step 5: Calculate Rotation Matrix Components

The rotation matrix components are \(\cos \theta\) and \(\sin \theta\). Find \(\sin 2\theta = -\frac{24}{25}\), \(\cos 2\theta = \frac{7}{25}\). Use the identities \(\sin\theta = \sqrt{\frac{1-\cos 2\theta}{2}}\) and \(\cos\theta = \sqrt{\frac{1+\cos 2\theta}{2}}\) with appropriate signs.

Step 6: Apply Rotation to Eliminate xy-Term

Substitute \(x = x'\cos\theta - y'\sin\theta\) and \(y = x'\sin\theta + y'\cos\theta\) into the original equation to eliminate the \(xy\)-term, expressing the equation in terms of \(x'\) and \(y'\). Simplify to get the equation without the \(xy\)-term.

Step 7: Sketch the Hyperbola

Sketch the hyperbola by first finding the new principal axes from the rotated equation. Identify the vertices and asymptotes from the transformed equation and graph accordingly, noting that it is a standard hyperbola graph.

Key Concepts

Discriminant in Conics

In conic sections, the discriminant is a fundamental tool used to determine the type of conic represented by a quadratic equation of the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). The discriminant formula is given by:

• \(D = B^2 - 4AC\)

By plugging in the coefficients, we can classify the conic:

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

For the expression \(11x^2 - 24xy + 4y^2 + 20 = 0\), the coefficients \(A\), \(B\), and \(C\) are 11, -24, and 4, respectively. Calculating the discriminant gives \(D = (-24)^2 - 4 \times 11 \times 4 = 400\), which is greater than 0, confirming that the graph represents a hyperbola. Understanding this logical use of \(D\) helps in swiftly identifying the conic type without graphing.

Rotation of Axes

Eliminating the \(xy\)-term in a conic section equation can simplify the graphing process. This is done using the rotation of axes. The rotation of axes works by transforming the coordinate system such that the mixed term \(xy\) disappears, leaving a simpler equation.
The angle \(\theta\) for rotation is found by first calculating \(\tan 2\theta\) using the formula:

• \(\tan 2\theta = \frac{B}{A - C}\)

Determine the angle \(2\theta\) from this ratio and then derive \(\theta\). Once \(\theta\) is known, we compute \(\cos \theta\) and \(\sin \theta\) using trigonometric identities. In this case:

• \(\sin 2\theta = -\frac{24}{25}\)
• \(\cos 2\theta = \frac{7}{25}\)

From these, \(\cos \theta = \sqrt{\frac{1 + \cos 2\theta}{2}}\) and \(\sin \theta = \sqrt{\frac{1 - \cos 2\theta}{2}}\). With these values, substitute the new coordinates \(x' = x\cos\theta - y\sin\theta\) and \(y' = x\sin\theta + y\cos\theta\) to express the equation in terms of \(x'\) and \(y'\) and eliminate the \(xy\)-term.

Hyperbola Classification

Classifying a hyperbola involves recognizing its standard form and characteristics. A hyperbola is defined mathematically by its eccentricity being greater than 1.
In standard form, the equation is either oriented horizontally or vertically:

• For a horizontal hyperbola: \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\).
• For a vertical hyperbola: \(\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1\).

Here, \((h, k)\) is the center of the hyperbola, \("a"\) and \("b"\) represent the distances from the center to the vertices and the co-vertices, respectively.
To sketch a hyperbola:

• Identify the center, \((h, k)\).
• Determine "a" and "b" to find the vertices and co-vertices.
• Draw asymptotes which pass through the center and guide the curvature.
• Sketch the two branches of the hyperbola opening according to the orientation (horizontally or vertically).

Applying these rules to the rotated form of the hyperbola equation allows us to graph it accurately.