Problem 9 - Contemporary Precalculus

Question

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$4 y^{2}-x^{2}+6 x-24 y+11=0$$

Step-by-Step Solution

Answer

Answer: The given equation represents a hyperbola, and a suitable viewing window is \( -20 \le x \le 20, \ -20 \le y \le 20\).

Step 1: Rewrite the given equation in the general conic form

We can rewrite the given equation as: $$-x^2 + 4y^2 + 6x - 24y + 11 = 0$$ In the general conic equation (\(Ax^2 + 2Bxy + Cy^2 + Dx + Ey + F = 0\)), the coefficients are: A = -1, B = 0, C = 4, D = 6, E = -24, and F = 11

Step 2: Calculate the discriminant and identify the conic section

Using the definition of the discriminant \(D = A^2 - 4BC\), we calculate the value: $$D = (-1)^2 - 4 \cdot 0 \cdot 4 = 1$$ Since D > 0, we can conclude that the conic section is a hyperbola.

Step 3: Select the viewing window

As the conic section is a hyperbola, we can experiment with different viewing windows to find one that will show the complete graph. One possible viewing window can be: $$ -20 \le x \le 20, \ -20 \le y \le 20$$ To conclude, the given equation represents a hyperbola, and a suitable viewing window is \( -20 \le x \le 20, \ -20 \le y \le 20\).

Key Concepts

Conic Sections

Conic sections are shapes created by intersecting a plane with a double-napped cone. Depending on the angle and position of this intersection, different conic shapes can be formed. These include the circle, ellipse, parabola, and hyperbola.

• A circle is formed when the intersecting plane is perpendicular to the cone's axis.
• An ellipse occurs when the plane cuts through the cone at an angle less than that of the cone.
• A parabola is the result of the plane being parallel to the slant height of the cone.
• A hyperbola results from the plane cutting through both nappes of the cone.

Understanding conic sections is essential in precalculus as they appear frequently in many mathematical contexts and applications, ranging from physics problems to engineering tasks.

Discriminant

The discriminant is a valuable tool for identifying the type of conic section represented by a given quadratic equation. It helps determine whether the graph of the equation will be a circle, an ellipse, a parabola, or a hyperbola.
The discriminant for the general conic section equation, \(Ax^2 + 2Bxy + Cy^2 + Dx + Ey + F = 0\), is given by the formula:\[ D = A^2 - 4BC \]

• If \(D = 0\), the equation represents a parabola.
• If \(D < 0\), the conic section is an ellipse or a circle.
• If \(D > 0\), the equation describes a hyperbola.

In our original exercise, calculating the discriminant revealed \(D = 1\), indicating a hyperbola because \(D\) is greater than zero.

Hyperbola

A hyperbola is one of the four fundamental types of conic sections, characterized by two separate curves or branches that mirror each other. It's important in precalculus due to its unique properties and applications.
Key characteristics of hyperbolas include:

• They consist of two disconnected curves that can open either horizontally or vertically.
• The standard form of a hyperbola's equation is \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) for horizontally-opened, and \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\) for vertically-opened curves.
• Each branch of the hyperbola gets closer to, but never actually touches, two asymptotes.

In the original problem, the equation \(-x^2 + 4y^2 + 6x - 24y + 11 = 0\) is identified as a hyperbola, which means it will be represented by two separate branches, and visualizing it requires choosing an appropriate range for both axes.