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Magazine Chapter 10: Problem 46
Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$9 x^{2}-y^{2}=10-2 y$$
Short Answer
Expert verified
The equation represents a hyperbola.
Step by step solution
01
Rearrange the Equation
First, we need to rearrange the given equation to a standard form. The given equation is: \[ 9x^2 - y^2 = 10 - 2y \]Add \(-2y\) to both sides to rearrange terms:\[ 9x^2 - y^2 + 2y = 10 \].
02
Complete the Square for y
To recognize the specific conic section, let's complete the square in terms of \(y\). The expression involving \(y\) is \( -y^2 + 2y \).Factor out the negative sign from the terms:\[ -(y^2 - 2y) \]To complete the square, take half of the coefficient of \(y\), which is \(-2\), divide it by 2 to get \(-1\), and square it to get \(1\). Add and subtract \(1\) inside the parenthesis:\[ -(y^2 - 2y + 1 - 1) \]This simplifies to:\[ -(y - 1)^2 + 1 \].Substitute back into the equation:\[ 9x^2 - (y - 1)^2 + 1 = 10 \].
03
Simplify the Equation
Subtract \(1\) from both sides to simplify:\[ 9x^2 - (y - 1)^2 = 9 \].
04
Identify the Conic Section
The equation \(9x^2 - (y-1)^2 = 9\) resembles the standard form of a hyperbola. The characteristic form of a hyperbola is:\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \].Divide the entire equation by 9 to match the standard hyperbola form:\[ \frac{x^2}{1} - \frac{(y-1)^2}{9} = 1 \].Since the equation is in the form where subtractive terms indicate a hyperbola, it clearly represents a hyperbola.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Hyperbola
A hyperbola is a type of conic section that appears when a plane intersects both parts of a double cone. A key feature of hyperbolas is that they have two distinct branches, making their graph differ from ellipses or circles. In particular, a hyperbola is characterized by a "difference" between the distances to two fixed points called foci. This crucial property is reflected in its standard form equation, which involves a subtraction.
- The basic standard equation for a hyperbola is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or the flipped form \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \).
- The minus sign and the two squares are highly noticeable in identifying a standard hyperbola.
- In a hyperbola, the transverse axis joins the vertices, cutting through the center, while the conjugate axis is perpendicular to it.
Completing the Square
Completing the square is a useful algebraic method for simplifying quadratic expressions or equations, enabling easier graphing and identification of conic sections like hyperbolas. This technique involves converting a quadratic expression into a perfect square trinomial:
- Consider a simple quadratic expression like \( x^2 + bx \).
- Take half the coefficient of \( x \), square it, and add and subtract it to the equation: \( x^2 + bx + \left( \frac{b}{2} \right)^2 - \left( \frac{b}{2} \right)^2 \).
- This forms a perfect square: \( \left( x + \frac{b}{2} \right)^2 - \left( \frac{b}{2} \right)^2 \).
Equation Rearrangement
Equation rearrangement is an important skill in mathematics that involves altering the structure of an equation without changing its solutions. This technique is crucial for transforming equations into a standard, recognizable form.
- First step involves moving terms around or combining like terms, usually aiming to isolate variables on one side.
- Rearranging may include operations like addition, subtraction, or factoring.
Standard Form of Conic Sections
The standard or canonical form of conic sections is a simpler representation of the various types of conic figures like circles, ellipses, parabolas, and hyperbolas. Each form makes the respective conic section easily identifiable and graphable.
- For hyperbolas, the standard form is \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \) or \( \frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1 \), where \((h, k)\) represents the center.
- Recognizing these forms allows for efficient graphing and understanding of the geometric properties involved.
- The "subtracting" nature of the squared terms is unique to hyperbolas among conics, while others might involve adding (ellipses, circles) or single variables (parabolas).
