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

Step-by-Step Solution

Answer

(a) The graph is a hyperbola. (b) Use rotation to eliminate the \(xy\)-term. (c) Sketch the hyperbola.

Step 1: Identifying the Discriminant

To determine the type of conic section described by the quadratic equation, we first calculate the discriminant. The discriminant for a general conic equation of the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\) is given by \(B^2 - 4AC\). For the given equation \(11x^2 - 24xy + 4y^2 + 20 = 0\), identify \(A = 11\), \(B = -24\), and \(C = 4\). Then compute the discriminant: \((-24)^2 - 4(11)(4)\).

Step 2: Calculating the Discriminant

Now compute \((-24)^2 - 4 \times 11 \times 4\). First, find \((-24)^2 = 576\). Next, calculate \(4 \times 11 \times 4 = 176\). Subtract \(176\) from \(576\) to get \(400\). Since the discriminant \(400\) is greater than zero, the conic section is a hyperbola.

Step 3: Eliminating the \(xy\)-Term via Rotation

To eliminate the \(xy\)-term, we need to perform a rotation of axes. The angle \(\theta\) for the rotation is given by \(\cot(2\theta) = \frac{A-C}{B}\). Plugging in the values, \(\cot(2\theta) = \frac{11 - 4}{-24} = \frac{7}{-24}\). Solve \(\theta\) using this equation which involves using the inverse cotangent function.

Step 4: Finding the Angle of Rotation

Solve for \(\theta\) such that \(\tan(2\theta) = \frac{-24}{7}\). Use the identity \(\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}\) to express \(\tan(\theta)\). It results in complex algebraic manipulations or using a calculator to find \(\theta\) approximately.

Step 5: Rewrite the Equation After Rotation

Use the angle found to substitute \(x = x'\cos(\theta) - y'\sin(\theta)\) and \(y = x'\sin(\theta) + y'\cos(\theta)\) into the original equation. After substitution and simplification, the new equation should be of the form \(A'x'^2 + C'y'^2 + D'x' + E'y' + F' = 0\) with no \(x'y'\)-term.

Step 6: Sketching the Graph

Once the rotated equation has no \(x'y'\) term, determine the new conic's center, axes, and sketches. Since it's a hyperbola, identify the vertices and asymptotes and sketch the hyperbola on the \(x'y'\)-plane.

Key Concepts

Discriminant of Conic Sections

The discriminant is a crucial part of understanding conic sections. It helps us differentiate between different types of conics such as parabolas, ellipses, and hyperbolas. The general form of a conic section equation is given by:\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]To find the discriminant, we use the formula \( B^2 - 4AC \). The values of \( A \), \( B \), and \( C \) are coefficients from the equation. Here's how the discriminant determines the type of conic:

• If \( B^2 - 4AC = 0 \), it's a parabola.
• If \( B^2 - 4AC > 0 \), it's a hyperbola.
• If \( B^2 - 4AC < 0 \), it's an ellipse.

For the exercise's equation \( 11x^2 - 24xy + 4y^2 + 20 = 0 \), we identified \( A = 11 \), \( B = -24 \), and \( C = 4 \). Plug these into the discriminant formula to get \((-24)^2 - 4 \times 11 \times 4 = 400\).
This positive result confirms that our conic section is a hyperbola.

Rotation of Axes

To simplify the equation of a conic section, especially when trying to remove the \(xy\)-term, the rotation of axes technique is employed. The general idea is to rotate the coordinate system to eliminate this term, making the equation easier to analyze.The angle of rotation, \(\theta\), is determined by the formula:\[ \cot(2\theta) = \frac{A - C}{B} \]For our given equation, substituting in the values we find:\[ \cot(2\theta) = \frac{11 - 4}{-24} = \frac{7}{-24} \]Solve for \( \theta \) by calculating the inverse cotangent. Numerical methods or a calculator make this task smoother.
After determining \( \theta \), the substitution:

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

This substitution converts the original equation into one that lacks an \(x'y'\)-term, thus providing a cleaner form to analyze the conic.

Hyperbola

A hyperbola is a conic section often identified by its characteristic shape with two disconnected curves. This shape appears prominently when the discriminant \(B^2 - 4AC > 0\) for a given equation.Key features of a hyperbola include:

• Two branches or "wings" opening opposite directions.
• A pair of asymptotes that the branches approach but never touch.
• A center, vertices, and foci that help define its shape.

To graph a hyperbola, start by locating its center and determining the distances to the vertices along the transverse axis. From here, sketch the asymptotes by drawing perpendicular lines intersecting at the center and use these guides to draw the hyperbola's curves.
A hyperbola's equation can be written in a more familiar form as:\[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \]If the conic has undergone rotation, adjust the coordinates using the rotation angle to fit this standard form.
Understanding hyperbolas in relation to other conic sections broadens the ability to tackle real-world applications, like satellite dish design and radiation patterns.