Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(x^{2}-6 x-2 y+7=0\)

Step-by-Step Solution

Answer

The given equation represents a Parabola.

Step 1: Identify the General Form of the Equation

The equation \(x^{2}-6x-2y+7=0\) given appears to be in the form of \(Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0\), which is a general form for a conic section. Here, \(B=0\) and \(C=0\) meaning there is no \(y^{2}\) term and no term having both x and y.

Step 2: Determine the Type of the Conic Section

Since there is only one squared term, this means that the equation represents a parabola. In general, if the equation of the conic section only has one squared term, it is either a parabola. Here, that is the case.

Key Concepts

Parabola

A parabola is a unique and interesting type of curve found in mathematics. It is defined as the set of all points that are equidistant from a point called the "focus" and a line called the "directrix." Parabolas have a characteristic u-shape and can open up, down, left, or right depending on their equation.
In their simplest form, parabolas can be described by the quadratic equation \(y = ax^2 + bx + c\). However, as seen in this exercise, we may encounter a case like \(x^2 - 6x - 2y + 7 = 0\) that also represents a parabola.
Here are some helpful ways to recognize a parabola:

• It has either an \(x^2\) or \(y^2\) term, but not both.
• The equation can often be rearranged to highlight its parabola form, such as completing the square.
• A parabola may also be shifted, rotated, or reflected, depending on coefficients and constants in the equation.

The exercise shows there is one squared term, \(x^2\), thus confirming the equation is a parabola. Understanding the basic features of a parabola will help you identify it even in its hidden forms.

Equation Classification

Classification of equations is a foundational skill in algebra and geometry. It helps in identifying the type of conic section an equation represents. Conic sections include parabolas, circles, ellipses, and hyperbolas. Recognizing these types can be made easier when equations are presented in their standard or general form.
For classification, look closely at the squared terms:

• If there is only one squared term (either \(x^2\) or \(y^2\), not both), the equation usually represents a parabola.
• If both \(x^2\) and \(y^2\) appear with the same coefficients, it is a circle.
• An ellipse will have \(x^2\) and \(y^2\) terms with different coefficients but the same sign.
• A hyperbola will have both \(x^2\) and \(y^2\) with opposite signs.

By recognizing these patterns, such as identifying a single squared term in this exercise, you can quickly classify the equation into the correct conic section.

General Form of a Conic Section

The general form of a conic section is a versatile equation that represents different types of curves, including ellipses, parabolas, circles, and hyperbolas. The general equation is written as:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
Each letter in this formula represents a coefficient that changes the shape and position of the conic section.
In our exercise, the equation \(x^2 - 6x - 2y + 7 = 0\) can be mapped into the general form, where:

• \(A = 1\), indicating the presence of \(x^2\)
• \(B = 0\), meaning there is no mixed \(xy\) term
• \(C = 0\), indicating there is no \(y^2\)

This configuration with only one squared term signifies the presence of a parabola. Each conic section has its own specific characteristics derived from these coefficients. Recognizing them allows for the categorization and further analysis of the curve's properties.