Problem 52 - Calculus Single Variable

Question

Which of the equations are circles? Which are not? Give precise reasons for your answers. \(x^{2}-y^{2}+6 x=-2 y^{2}-7\)

Step-by-Step Solution

Answer

The equation is not a circle; it is likely an ellipse.

Step 1: 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 \[ x^2 + y^2 + 6x + y^2 = -7 \].

Step 2: 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.

Step 3: 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.

Key Concepts

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: \( (x - h)^2 + (y - k)^2 = r^2 \).

This equation clearly highlights the symmetry in both the \( x \) and \( y \) dimensions, as the coefficients of \( x^2 \) and \( y^2 \) are equal. Whenever these coefficients are unequal, the figure represented by the equation is no longer a circle. Such a distinction is vital when classifying shapes derived from algebraic equations.

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.

When the coefficients of \( x^2 \) and \( y^2 \) in the equation are unequal, it signifies an ellipse instead of a circle. The given equation, after simplification in the original exercise, exemplifies this property with terms like \( x^2 + 2y^2 \), showcasing a different scaling along the \( y \) direction compared to the \( x \) direction.

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 \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\).

The standard form is especially helpful in transforming and simplifying equations to check their geometric properties. Initially, the given equation \( x^2 - y^2 + 6x = -2y^2 - 7 \) was not clear about its geometric form. By simplifying and comparing it to standard equations, it's much easier to identify whether the equation represents a circle or an ellipse.

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.

Analyzing coefficients allows us to predict how figures behave geometrically. In our context:

• 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.

This understanding aids in differentiating and categorizing second-degree equations more effectively.