Problem 59 - Precalculus with Limits

Question

In Exercises 57-72, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(4x^2-y^2-4x-3=0\)

Step-by-Step Solution

Answer

The graph of the given equation is a Hyperbola, with the center at (0.5,0), and a=1, b=2.

Step 1: Identify the general form

The given equation \(4x^2-y^2-4x-3=0\) is a form of the general equation for a conic section \(Ax^2 + By^2 + Cx + Dy + E = 0\).

Step 2: Re-arrange and complete the square

Re-arrange the given equation to group variables: \(4x^2-4x - y^2 = 3\). Now, complete the square for both x and y variables. For x-terms : \(4(x^2 - x) = 4[(x-0.5)^2 - 0.25]\), adding and subtracting 0.25 within the bracket. By simplifying we get \(4(x-0.5)^2 -1\). So, the equation becomes \(4(x-0.5)^2 - y^2 = 4\). Dividing by 4, this gets to \((x-0.5)^2 - \frac {y^2}{4} = 1\).

Step 3: Identify the conic section

This is the standard form of a hyperbola \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\), where a, b are the constants and h, k are the coordinates of the center. Here, the center of the hyperbola is at (0.5,0), a=1, b=2.

Key Concepts

Hyperbola

A hyperbola is a fascinating conic section that consists of two separate curves, called branches. These curves are symmetric with respect to both the axes and the center. Hyperbolas have unique properties compared to other conic sections like ellipses, parabolas, and circles. The key feature of a hyperbola is that the absolute difference between the distances from any point on the hyperbola to the two fixed points called foci is constant.

For the standard form of a hyperbola: - \[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \- \] The branches open left and right.- Where \( (h, k) \) is the center, \( a^2 \) and \( b^2 \) are the squared lengths of the semi-major and semi-minor axes, respectively.- The transverse axis is horizontal, and the foci are located at a distance of \( c \) from the center, where \( c^2 = a^2 + b^2 \).- Remember, if the \( y^2 \) term is positive, then the hyperbola opens up and down instead of left and right.

Understanding hyperbolas is essential as they occur naturally in various fields including physics, engineering, and astronomy, representing paths of comets or the shape of certain satellite dishes.

Equation Classification

Equation classification helps identify which type of conic section an equation represents. Conic sections include circles, ellipses, parabolas, and hyperbolas, and each has its specific standard form.

The general form of a conic section's equation is - \[ Ax^2 + By^2 + Cx + Dy + E = 0 \- \] where the coefficients \( A \) and \( B \) determine the type based on their values:

• If \( A = B \) and both are nonzero, the equation represents a circle.
• If \( A eq B \) but both are positive, it represents an ellipse.
• If either \( A \) or \( B \) is zero, it indicates a parabola.
• If \( A \) and \( B \) have opposite signs, like \( A > 0 \) and \( B < 0 \) or vice versa, the equation forms a hyperbola.

Classifying the graph of an equation helps us understand its geometry and structure, simplifying tasks like graphing or analyzing the conic section's properties.

Completing the Square

Completing the square is a mathematical technique used to simplify polynomial equations, making it easier to identify their structure, such as locating the vertex of a parabola or converting conic equations into their standard forms.

To complete the square:1. Focus on the quadratic terms. For example, let's use \( x^2 - 4x \) from the given equation.2. Take the linear term's coefficient (in this case, -4), halve it to get -2, and square it to get 4. Add and subtract this square within the quadratic expression: - \( x^2 - 4x + 4 - 4 = (x-2)^2 - 4 \)3. Replace the original quadratic expression with its squared form: - For \( 4(x^2-x) \), this would become \( 4[(x-0.5)^2 - 0.25] \) simplified as \( 4(x-0.5)^2 - 1 \).4. This technique helps transform the equation into a more recognizable and manageable form, clarifying whether it represents a conic such as a hyperbola, parabola, or ellipse.

Mastering completing the square is valuable for solving and understanding quadratic equations, particularly in graphing and analyzing geometric properties of conics.