Problem 29 - Algebra and Trigonometry

Question

(a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$6 x^{2}+10 x y+3 y^{2}-6 y=36$$

Step-by-Step Solution

Answer

The conic is a hyperbola, confirmed by graphing.

Step 1: Identify the Discriminant

The discriminant of a conic section equation in the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\) is given by the formula \(D = B^2 - 4AC\). For this exercise, we have \(A = 6\), \(B = 10\), \(C = 3\). We compute the discriminant as follows: \(D = 10^2 - 4(6)(3) = 100 - 72 = 28\).

Step 2: Determine the Type of Conic Based on Discriminant

The discriminant \(D = 28\) is greater than zero. When \(D > 0\), the conic is a hyperbola.

Step 3: Graph the Conic Section

Using a graphing device or software, plot the equation \(6x^2 + 10xy + 3y^2 - 6y = 36\). The graph should confirm the shape and characteristics of a hyperbola, solidifying the identification made using the discriminant.

Key Concepts

Discriminant of a Conic

The discriminant is a powerful tool used to identify the type of conic section represented by a second-degree equation. Conic sections include circles, parabolas, ellipses, and hyperbolas. Knowing which conic a given equation represents helps us understand its geometric properties and behavior.
In the general conic equation form, \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the discriminant, denoted as \(D\), is calculated using the formula \(D = B^2 - 4AC\). This discriminant helps classify the conic section:

• If \(D < 0\), the conic is an ellipse. A special case when \(A = C\) and \(B = 0\), it represents a circle.
• If \(D = 0\), it is a parabola.
• If \(D > 0\), the conic is a hyperbola.

Applying this to our equation \(6x^2 + 10xy + 3y^2 - 6y = 36\), we find \(A = 6\), \(B = 10\), and \(C = 3\), leading to a discriminant \(D = 28\), confirming the equation represents a hyperbola.

Hyperbola

A hyperbola is a type of conic section characterized by two separate, symmetrical curves. These curves are mirror images of each other and are often associated with phenomena that involve accelerating forces or reflectivity, such as satellite dishes and hyperbolic trajectories in space.
Hyperbolas have a central point, known as the center, and two foci. These foci are pivotal for defining the hyperbola because any point on a hyperbola's curve maintains a constant difference in distances to each focus.

• The equation of a standard hyperbola centered at the origin with a horizontal transverse axis is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
• If it's centered at the origin with a vertical axis, it's \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).

In the given equation from the exercise, the non-standard form \(6x^2 + 10xy + 3y^2 - 6y = 36\) still reveals a hyperbola due to the positive discriminant. Graphing the equation helps visually confirm its identity as a hyperbola, showing its two characteristic curves.

Graphing Conic Sections

Graphing conic sections is an excellent way to understand their unique geometric properties. Each type of conic section—circle, ellipse, parabola, and hyperbola—has distinct graph shapes and characteristics.
To graph a conic section, you can use graphing software or a graphing device to input the equation directly. They accurately plot the equation based on the defined algebraic terms, unveiling the shape and position of the conic.
When graphing a hyperbola, it is important to:

• Identify the center, vertices, and foci, using these features to draw the asymptotes that guide the curves of the hyperbola.
• Ensure the equation is rearranged in standard form if necessary, which may require completing the square or other algebraic manipulations.

In the exercise, graphing \(6x^2 + 10xy + 3y^2 - 6y = 36\) confirms that it forms a hyperbola. It further helps learners visually verify the discriminant's conclusion by examining the symmetrical branches of the hyperbola presented on the graph.