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Chapter 10: Problem 42

Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola. $$ 2 x^{2}+12 x+18-y^{2}=3\left(2-y^{2}\right)+4 y $$

Short Answer

Expert verified
The graph is a hyperbola.

Step by step solution

01

Simplify both sides separately

Start by simplifying both sides of the given equation \( 2x^2 + 12x + 18 - y^2 = 3(2 - y^2) + 4y \). The left side is already expanded. For the right side, distribute the 3 in the term \( 3(2-y^2) \). This results in \( 6 - 3y^2 + 4y \).
02

Rearrange terms to detect the conic section

Now, write both sides in one equation: \( 2x^2 + 12x + 18 - y^2 = 6 - 3y^2 + 4y \). Move all terms to one side to begin analyzing the conic type: \( 2x^2 + 12x + y^2 - 3y^2 - 4y + 18 - 6 = 0 \). Simplifying the equation, you get: \( 2x^2 + 12x - 2y^2 - 4y + 12 = 0 \).
03

Identify the conic based on coefficients

Identify the type of conic section based on the coefficients. Notice the squared terms: there is a \( x^2 \) term and a \( y^2 \) term with opposite signs (\( 2x^2 - 2y^2 \)). This indicates the presence of both positive and negative terms indicates the conic is a hyperbola.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parabola
A parabola is a unique and simple conic section that you can recognize by its U-shaped curve. The standard form of a parabolic equation is either \( y = ax^2 + bx + c \) or \( x = ay^2 + by + c \). A parabola has a single axis of symmetry and a single point known as the vertex, which represents the highest or lowest point on the graph depending on its orientation. There are several key characteristics:
  • Vertex: The turning point of the parabola.
  • Focus: A point inside the parabola that helps define its shape.
  • Directrix: A line outside the parabola that is equidistant from the vertex as the focus is.
  • Axis of Symmetry: A line that divides the parabola into two mirror images.
Remember, for a parabola, only one variable is squared. This simplicity makes it a frequent starting point when studying conics.
Circle
Circles are perhaps the most well-known conic sections. They are defined as the set of all points in a plane that are equidistant from a given point called the center. The equation for a circle can be written in standard form as \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) represents the center, and \(r\) is the radius. Circles have several key properties:
  • Center: The middle point from which all points on the circle are equidistant.
  • Radius: The distance from any point on the circle to its center.
  • Diameter: A straight line that passes through the center of the circle, connecting two points on its edge, and is twice the radius.
  • Circumference: The perimeter or boundary length of the circle.
Unlike other conics, circles have no squared term with a different coefficient from the others. This symmetrical quality makes them instantly recognizable.
Ellipse
An ellipse is an oblong shape that resembles a stretched or squashed circle. It is represented by the standard equation \( \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \), where \( (h, k) \) is the center, \( a \) is the semi-major axis, and \( b \) is the semi-minor axis. Key features of ellipses include:
  • Foci: Two points inside the ellipse that help to define the shape.
  • Major Axis: The longest diameter of the ellipse that passes through its center.
  • Minor Axis: The shortest diameter perpendicular to the major axis.
  • Eccentricity: A measure of how elongated the ellipse is.
The sum of distances from any point on the ellipse to the two foci is constant. This property makes ellipses the focus in optics and astronomy.
Hyperbola
A hyperbola consists of two symmetrical open curves. It is represented by the equation \( \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \) or its alternate form when \( y^2 \) precedes the \( x^2 \). Hyperbolas have two branches that mirror each other, and their key characteristics include:
  • Foci: Two points from which distances are measured in defining the curve.
  • Vertices: The points where each branch of the hyperbola is closest to its center.
  • Asymptotes: Lines that the branches approach but never touch. They provide a framework for the curve.
  • Center: The midpoint between the vertices (and foci).
A telling feature of hyperbolas is the presence of terms with different coefficients and one positive and one negative square term. They often model real-life scenarios such as sound waves and orbit paths.

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Most popular questions from this chapter

Find the exact solution(s) of each system of equations. $$ \begin{array}{l}{y^{2}+x^{2}=25} \\ {y^{2}+9 x^{2}=25}\end{array} $$

Find the exact solution(s) of each system of equations. $$ \begin{array}{l}{x^{2}+y^{2}=36} \\ {y=x+2}\end{array} $$

Explain how hyperbolas and parabolas are different. Include differences in the graphs of hyperbolas and parabolas and differences in the reflective properties of hyperbolas and parabolas.

ACT/SAT How many solutions does the system of equations \(\frac{x^{2}}{5^{2}}-\frac{y^{2}}{3^{2}}=1\) and \((x-3)^{2}+y^{2}=9\) have? A 0 B 1 C 2 D 4

Rockers Two rockets are launched at the same time, but from different heights. The height \(y\) in feet of one recket after \(t\) seconds is given by \(y=-16 t^{2}+150 t+5 .\) The height of the other rocket is given by \(y=-16 t^{2}+\) 160\(t\) . After how many seconds are the rockets at the same height?
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