Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, determine which conic section is represented based on the given equation. \(4 x^{2}-y^{2}+8 x-1=0\)

Step-by-Step Solution

Answer

The equation represents a hyperbola.

Step 1: Rearrange the Equation

Given the equation is \(4x^{2} - y^{2} + 8x - 1 = 0\). We will rearrange it to group the \(x\) terms together and move the constant to the other side: \(4x^{2} + 8x - y^{2} = 1\).

Step 2: Identify the Standard Form

The equation resembles the general form of a conic section, which is either an ellipse, hyperbola, or parabola. When rearranging, we need to figure out whether the coefficients of \(x^2\) and \(y^2\) have opposite signs, which indicates it's a hyperbola.

Step 3: Recognize Coefficients and Signs

In the rearranged equation \(4x^{2} + 8x - y^{2} = 1\): The coefficient of \(x^2\) is \(4\) and \(y^2\) is \(-1\). Since one is positive and the other is negative, this indicates a hyperbola.

Key Concepts

Hyperbola

A hyperbola is one of the four types of conic sections, and it has unique features that differentiate it from other conics like circles, ellipses, and parabolas. The defining property of a hyperbola is that the differences in distances from any point on the curve to two fixed points (the foci) are constant. Its shape resembles two open curves, or branches, facing away from each other, and it can have either horizontal or vertical orientations. Here's what's interesting about hyperbolas:

• There are always two branches in a hyperbola.
• Hyperbolas have two axes of symmetry, which are two lines that divide the figure into mirror-image halves.
• The vertices of a hyperbola are the points where the branches come closest to each other.

Hyperbolas are often used to model real-world scenarios involving direct and indirect movements, such as the paths of objects in space.

Equations of Conic Sections

Conic sections can be described using different types of equations, highlighting their geometric properties. The main types of conic sections are the circle, ellipse, parabola, and hyperbola. Each has equations that define its specific form and characteristics.Key features of conic section equations include:

• Circles: The equation has the general form \((x-h)^2 + (y-k)^2 = r^2\), where \( (h, k) \) is the center and \( r \) is the radius.
• Ellipses: Similar to circles but with different radii, expressed with \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \).
• Hyperbolas: Have the form \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \) or \(\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1 \), characterized by having opposite signs compared to ellipses.
• Parabolas: Generally represented by \((y-k) = a(x-h)^2\) or \((x-h) = a(y-k)^2\), indicating a single curved path.

Understanding these equations allows us to recognize which type of conic section is represented in problems and their unique properties. Recognizing opposite signs, like in the case of the hyperbola, helps to identify the conic section quickly.

Standard Form of Conics

The standard form of conic sections provides a simplified and structured way of expressing conic equations, which helps in identifying and graphing the type of conic section. These forms are derived from the general quadratic equation in two variables:\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]In the case of a hyperbola, breaking down the general quadratic equation into a standard form makes it easy to comprehend its geometry:

• The standard form for a
  • horizontal hyperbola: \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\)
  • vertical hyperbola: \(\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1\)
• For hyperbolas, distinguish between horizontal and vertical based on the placement and sign of terms.
• Completing the square is often used to rearrange conic equations into a more recognizable standard form.

Reordering and simplifying an equation to fit the standard form helps to clearly identify the center, vertices, and asymptotes of the hyperbola, making it easier to graph and understand.