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Chapter 10: Problem 14

Determine whether the graph of the equation will be a circle, a parabola, an ellipse, or a hyperbola. a. \(x^{2}+y^{2}=10\) b. \(9 y^{2}-16 x^{2}=144\) c. \(x=y^{2}-3 y+6\) d. \(4 x^{2}+25 y^{2}=100\)

Short Answer

Expert verified
a. Circle, b. Hyperbola, c. Parabola, d. Ellipse.

Step by step solution

01

Analyzing Equation a

The given equation is \(x^{2} + y^{2} = 10\). This equation fits the standard form of a circle, which is \(x^{2} + y^{2} = r^{2}\), where \(r\) is the radius. Therefore, this equation represents a circle.
02

Analyzing Equation b

The given equation is \(9y^{2} - 16x^{2} = 144\). This equation can be rewritten as \(\frac{9y^{2}}{144} - \frac{16x^{2}}{144} = 1\) or \(\frac{y^{2}}{16} - \frac{x^{2}}{9} = 1\), which fits the standard form of a hyperbola \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\). Therefore, this equation represents a hyperbola.
03

Analyzing Equation c

The given equation is \(x = y^{2} - 3y + 6\). This is not a quadratic in \(x\), but rather it is a quadratic in \(y\). It can be rewritten in the standard form \(y = a(y^{2} - 3y + 6)\), which is that of a parabola. Therefore, this equation represents a parabola.
04

Analyzing Equation d

The given equation is \(4x^{2} + 25y^{2} = 100\). Dividing through by 100 gives \(\frac{x^{2}}{25} + \frac{y^{2}}{4} = 1\), which matches the standard form of an ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\). Therefore, this equation represents an ellipse.

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Key Concepts

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

Circle Equation
A circle equation in its simplest form looks like \( x^{2} + y^{2} = r^{2} \), where \( r \) is the radius. The center of this circle is at the origin, \((0,0)\). If the terms involving \( x \) and \( y \) are squared and positive, and they have equal coefficients, the graph is a circle. The equation \( x^{2} + y^{2} = 10 \) follows this format. Here, the value of \( r^{2} \) is 10, indicating that the radius \( r \) is \( \sqrt{10} \). It's this simplicity and symmetry that makes circles easy to identify in mathematics. To identify a circle, look for:
  • Both \( x^{2} \) and \( y^{2} \) terms.
  • Both terms have the same coefficient (1 in this case).
  • The equation equals to a constant, which is \( r^{2} \).
Hyperbola Equation
Hyperbolas have a distinctive equation structure, which is \( \frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1 \) or \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \). The sign difference between terms indicates the characteristic open shape. For equation \( 9y^{2} - 16x^{2} = 144 \), divide through by 144 to convert it into standard form: \( \frac{y^{2}}{16} - \frac{x^{2}}{9} = 1 \). This shows a hyperbola opening vertically, as \( y^{2} \) has the positive term. Hyperbolas are composed of two branches, using the center and the axes to guide you in the graph.Key signs of a hyperbola are:
  • One \( x^{2} \) or \( y^{2} \) term is subtracted from the other.
  • Constructed from the division of terms by different values forming the fractions.
Parabola Equation
A parabola has a form that typically involves one variable being squared, such as \( y = ax^{2} + bx + c \) or \( x = ay^{2} + by + c \). Parabolas are unique in that they only curve in one direction, creating a U-shape or an inverted U-shape. For \( x = y^{2} - 3y + 6 \), it's quadratic in \( y \), making \( x \) dependent on \( y \). This gives the form \( y = a(y^{2} - 3y + 6) \), and highlights how parabolas can open sideways as well. They focus on a single axis of symmetry, moving either horizontally or vertically based on the squared variable.Characteristics of a parabola:
  • Only one variable is squared.
  • Opens in a single direction relying on its equation orientation.
Ellipse Equation
Ellipses appear as stretched circles and have the form \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \). This equation signals an ellipse, distinguished by the sum of squared terms creating an elongated shape. The key is that \( a^{2} \) and \( b^{2} \) are different, making it wider or taller. Equation \( 4x^{2} + 25y^{2} = 100 \) divides by 100 to line up with \( \frac{x^{2}}{25} + \frac{y^{2}}{4} = 1 \). This indicates an ellipse centered at the origin with lengths along the axes dictated by those denominators.To identify an ellipse, check for:
  • Both \( x^{2} \) and \( y^{2} \) terms added together.
  • Different values under \( x^{2} \) and \( y^{2} \) indicating its axes lengths and orientation.

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Most popular questions from this chapter

Solve each system of equations by graphing. See Example 1. $$ \left\\{\begin{array}{l} x^{2}+4 y^{2}=4 \\ x=2 y^{2}-2 \end{array}\right. $$

Solve each system of equations for real values of \(x\) and \(y.\) $$ \left\\{\begin{array}{l} x^{2}+y^{2}=4 \\ 9 x^{2}+y^{2}=9 \end{array}\right. $$

Write each equation in standard form, if it is not already so, and graph it. The problems include equations that describe circles, parabolas, ellipses, and hyperbolas. $$ \frac{x^{2}}{1}+\frac{y^{2}}{36}=1 $$

Solve each system of equations by substitution for real values of \(x\) and \(y.\) See Examples 2 and 3. $$ \left\\{\begin{array}{l} x^{2}-x-y=2 \\ 4 x-3 y=0 \end{array}\right. $$

Compare the graphs of \(\frac{x^{2}}{81}-\frac{y^{2}}{64}=1\) and \(\frac{y^{2}}{81}-\frac{x^{2}}{64}=1 .\) Do they have any similarities?
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