Problem 66 - Precalculus with Limits

Question

In Exercises 57-72, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(4y^2-2x^2-4y-8x-15=0\)

Step-by-Step Solution

Answer

The graph of the equation \(4y^2-2x^2-4y-8x-15=0\) is a hyperbola with a center at (2, 0.5), A=sqrt(2) and B=sqrt(2).

Step 1: Rearrange the equation

The first step is to rearrange the given equation by grouping the x terms and y terms together: \(2(2y^2-2y)+(-2x^2+8x)=-15\). This can be written as: \(2[(2y^2-2y)+1]-2[(x^2-4x)+4]=-15+2-8\) \nThis simplifies to: \(2[(y-0.5)^2]-2[(x-2)^2]=-21\)

Step 2: Identify the type

We identify conic sections by the coefficients of \(x^2\), \(y^2\) and the signs in their equations. A circle has both coefficients equal and addition, an ellipse has the coefficients different and addition, a parabola has only one squared variable, a hyperbola has both coefficients equal/different and substraction. The rearranged form of the equation is in the form of \(A[(y-h)^2]-B[(x-k)^2]=C\), where A, B, and C are constants. A > 0 and B > 0. So we can see that the rearranged equation is a hyperbola.

Step 3: Find the center, A and B

In the rearranged equation, the terms in the parentheses give the coordinates of the center of the hyperbola, obtained by setting \(y-0.5 = 0\) and \(x-2 = 0\). So the center of the hyperbola is (2, 0.5). Also, A and B can be found by taking the square root of the constants before corresponding parentheses. So, A = sqrt(2), B = sqrt(2).

Key Concepts

Hyperbola

Understanding the properties of a hyperbola helps to classify equations accurately. A hyperbola is a type of conic section formed when a plane intersects both nappes of a double cone. This results in two symmetric curves, open in opposite directions. Unlike ellipses, which are closed, hyperbolas extend infinitely.

• General equation: For a hyperbola centered at ((h, k)), the standard form is \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\) or \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\).
• Axes orientation: Depending on the leading positive term, the hyperbola can open horizontally or vertically.
• Asymptotes: The asymptotes of the hyperbola provide a boundary that the hyperbola approaches but never meets.

Understanding these features helps distinguish hyperbolas from other conic sections like ellipses or parabolas.

Graph Classification

Classifying graphs involves identifying which conic section an equation represents. Conic sections include circles, ellipses, parabolas, and hyperbolas.

• Circle: Shows equal coefficients for \(x^2\) and \(y^2\) with addition.
• Ellipse: Includes different coefficients for \(x^2\) and \(y^2\) with addition.
• Parabola: Consists of only one squared variable.
• Hyperbola: Features differing signs between the \(x^2\) and \(y^2\) terms.

Here, the presence of a negative sign between the \(x^2\) and \(y^2\) terms indicates that the equation represents a hyperbola. Recognizing these indicators is key to graph classification.

Equation Rearrangement

Rearranging equations is a crucial step in classifying conic sections. It involves isolating and organizing terms effectively.Start by moving constant terms and grouping similar terms, such as \(x\) and \(y\) terms, together.
For example, with the provided equation \(4y^2 - 2x^2 - 4y - 8x - 15 = 0\), we group and simplify:

• Combine the \(y\) and \(x\) terms: \(2(2y^2 - 2y) + (-2x^2 + 8x) = -15\).
• Complete the square for both \(x\) and \(y\) groups to simplify the expression.

Rearranging is not just about solving; it's about transforming the equation into a recognizable form, making it easier to classify.

Coordinate Geometry

Coordinate geometry offers a structured approach to understanding and plotting equations in the Cartesian plane. Utilizing coordinates simplifies the examination of algebraic equations and geometric figures.

• Center: For conic sections, identifying the center is critical. Here, the center of the hyperbola is derived from the terms inside the rearranged squares: ((2, 0.5)).
• Vertices and Focus: Use formulas involving \(a\) and \(b\) from the equation's standard form to find these points' coordinates.
• Axes: Recognize whether they are major, minor, transverse, or conjugate, depending on the conic section type.

Mastering these concepts in coordinate geometry enhances the ability to not only plot but also interpret various geometric shapes.