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Magazine Chapter 1: Problem 52
Which of the equations are circles? Which are not? Give precise reasons for your answers. \(x^{2}-y^{2}+6 x=-2 y^{2}-7\)
Short Answer
Expert verified
The equation is not a circle; it is likely an ellipse.
Step by step solution
01
Rewrite the Equation
Start by rewriting the given equation: \(x^2 - y^2 + 6x = -2y^2 - 7\). Move all terms to one side to simplify:\[ x^2 - y^2 + 6x + 2y^2 = -7 \]which can be rearranged as:\[ x^2 + (2 + 1)y^2 + 6x = -7 \] or \[ x^2 + y^2 + 6x + y^2 = -7 \].
02
Simplify and Identify the Terms
Now, let's simplify it to see if it matches the general equation of a circle. \[ x^2 + y^2 + 6x + y^2 = -7 \] Combine like terms: \[ x^2 + 2y^2 + 6x = -7 \] This is not in the form of a circle equation because of the \( 2y^2 \). The general form for a circle is \( x^2 + y^2 + Dx + Ey = F \). The term \( 2y^2 \) signifies different scaling in the \( y \) direction, which is characteristic of an ellipse rather than a circle.
03
State the Conclusion
The given equation \( x^2 - y^2 + 6x = -2y^2 - 7 \) simplifies to \( x^2 + 2y^2 + 6x = -7 \), which is not in the standard form for a circle equation due to the presence of \( 2y^2 \). Therefore, this equation does not represent a circle; it resembles the form of an ellipse due to the unequal coefficients of the \( x^2 \) and \( y^2 \) terms.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Equation of a Circle
The equation of a circle is an important foundational concept in geometry, representing a set of all points in a plane that are equidistant from a given point. This given point is known as the "center" of the circle. In its simplest form, the equation of a circle centered at the origin
- is represented as: \( x^2 + y^2 = r^2 \).
- Here, \( r \) represents the radius of the circle.
- The circle's center at any point \((h, k)\) modifies the equation to: \( (x - h)^2 + (y - k)^2 = r^2 \).
Ellipse
An ellipse is another type of conic section, which can be mistakenly assumed to be a circle if not correctly identified. Structurally, an ellipse is defined by the equation:
- \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \).
- This indicates a symmetric shape stretched differently along the \( x \) and \( y \) axes.
- The values \( a \) and \( b \) determine these stretches and are known as the semi-major and semi-minor axes, respectively.
Standard Form
In algebra, the "standard form" refers to a way of presenting mathematical equations to make them easy to understand and use. For conic sections like circles, the standard form helps identify and describe the curves accurately. For instance:
- A circle's standard form is \((x-h)^2 + (y-k)^2 = r^2\).
- An ellipse's standard form is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\).
Coefficients
Coefficients in algebraic equations play a significant role, especially in determining the nature of conic sections. They are numbers multiplied directly to variables, which influence the shape and projective properties of figures.
- In a circle, the coefficients beside \( x^2 \) and \( y^2 \) should be equal.
- If they differ, like \( x^2 \) and \( 2y^2 \), it indicates a stretch that is not uniform, pointing towards an ellipse.
- The original exercise highlights this principle when converting the equation to \( x^2 + 2y^2 + 6x = -7 \), signifying it cannot be a circle but matches an ellipse's properties.
