Problem 53 - Calculus Single Variable

Question

Which of the equations are circles? Which are not? Give precise reasons for your answers. \(x^{2}+2 y^{2}+x-5 y=7+3 x^{2}\)

Step-by-Step Solution

Answer

The equation is not a circle because the coefficients of \(x^{2}\) and \(y^{2}\) are not equal and there are additional linear terms.

Step 1: Write the Equation in a Standard Form

Start by writing the given equation in a more standard form by moving all terms to one side of the equation. To do this, subtract \(3x^{2}\) from both sides.The equation becomes:\[x^{2} - 3x^{2} + 2y^{2} + x - 5y - 7 = 0\]

Step 2: Simplify the Equation

Combine like terms in the equation, specifically the \(x^{2}\) terms:\[-2x^{2} + 2y^{2} + x - 5y - 7 = 0\]

Step 3: Identify the Characteristics of a Circle's Equation

An equation for a circle is typically of the form \[(x-h)^{2} + (y-k)^{2} = r^{2}\]where \(h\) and \(k\) represent the circle's center and \(r\) is its radius. In this form, the coefficients of \(x^2\) and \(y^2\) are equal. Also, there should be no other term combining \(x\) and \(y\).

Step 4: Analyze the Given Equation

Examine the simplified version of the equation:\[-2x^{2} + 2y^{2} + x - 5y - 7 = 0\]Notice the coefficients of \(x^{2}\) and \(y^{2}\) are not equal: \(-2\) and \(2\) respectively. This inequality means the graph of this equation cannot be a circle. Additionally, there are linear terms \(x\) and \(y\) that aren't part of a perfect square, which is inconsistent with the circle's standard form.

Step 5: Conclusion on the Circle's Identification

Since the coefficients of \(x^{2}\) and \(y^{2}\) are not equal, and because of extra linear terms, the given equation does not represent a circle.

Key Concepts

Standard Form of a Circle

To determine whether an equation represents a circle, it's essential to understand the standard form of a circle's equation. This form is \((x-h)^{2} + (y-k)^{2} = r^{2}\). Here, \(h\) and \(k\) identify the center of the circle, while \(r\) is the radius.
An equation meeting these criteria will ensure:

• Equal coefficients for \(x^{2}\) and \(y^{2}\)
• The absence of additional linear \(x\) and \(y\) terms

In our exercise, the given equation was \(-2x^{2} + 2y^{2} + x - 5y - 7 = 0\), which clearly doesn't meet the standard form requirements as the coefficients \(-2\) and \(2\) are not equal, and there are extra linear terms. Thus, it doesn't represent a circle.
Understanding the standard form is crucial because it simplifies the identification and graphing process for equations of circles, making it easier to solve problems related to circle geometry.

Quadratic Equations

Quadratics play a significant role in forming the base of many geometrical shapes. In the context of circles, quadratic equations are analyzed through their components: \(x^{2}\) and \(y^{2}\) terms. A true quadratic circle equation must have equal coefficients for these terms.
This balance between \(x^{2}\) and \(y^{2}\) results in a perfect square, an essential quality for describing a circle. However, not all quadratic equations comply, and in the exercise, the given equation displays an inequality between these coefficients. This indicates that the equation is not a simple circle but could represent other conic sections or a more complex graph.
Recognizing these specifics helps in distinguishing whether a quadratic equation fits within the circle category, which is a fundamental element for solving related mathematical problems.

Graphing

Graphing equations involves visually translating mathematical expressions into coordinate planes. For equations of circles, this means drawing a round shape that corresponds perfectly with the circle's formula.
The exercise's initial equation, when simplified, did not lead to a typical circle-graph due to unequal \(x^{2}\) and \(y^{2}\) coefficients and extra terms. If an equation deviates from the standard circle form, its graph will demonstrate that by diverging from the expected circular path.
Graphing is an essential tool for verifying theoretical mathematical predictions. By sketching an equation on a graph, students can visually check their solutions against the expected shapes, solidifying their understanding of concepts like circle geometry and its graphical representation.

Circle Geometry

Circle geometry focuses on understanding the properties and character of circles. This involves recognizing the elements like the center, radius, and the symmetry inherent in circular shapes.
In the given exercise, understanding these geometric principles allows students to assess whether an equation truly represents a circle. Checking if the equation aligns with circle properties, like having equal \(x^{2}\) and \(y^{2}\) coefficients, helps in employing logical reasoning to solve the exercise.
Recognizing circles and their characteristics in equations is not just about shape, but about symmetry and balance. This aspect is essential in geometric problem-solving and further explores how these properties apply in more extensive mathematical contexts.