Problem 23 - A Graphical Approach to College Algebra

Question

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$6 x^{2}-12 x+6 y^{2}-18 y+25=0$$

Step-by-Step Solution

Answer

The equation represents a circle.

Step 1: Recognize the General Form

We begin by recognizing that this equation is in a general quadratic form: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]In the given equation, it is: \[ 6x^2 - 12x + 6y^2 - 18y + 25 = 0 \] From this, we identify \( A=6 \), \( B=0 \), \( C=6 \), \( D=-12 \), \( E=-18 \), and \( F=25 \).

Step 2: Identify Conic Section Type

The type of conic section is determined by the coefficients \( A \), \( B \), and \( C \). Use the discriminant \( B^2 - 4AC \) to distinguish between ellipse, parabola, and hyperbola.Substitute the coefficients:\[ B^2 - 4AC = 0^2 - 4 \times 6 \times 6 = -144 \]Since \( B^2 - 4AC < 0 \), the conic is an ellipse or a circle.

Step 3: Check for Circle

An ellipse can further be identified as a circle if the coefficients \( A \) and \( C \) are equal and \( B = 0 \).Here, \( A = 6 \) and \( C = 6 \), and indeed \( B = 0 \). Therefore, the given equation is a circle.

Key Concepts

Equation of a Circle

The equation of a circle is a specific form of conic section. Circles are unique because they have constant radii from a central point. In mathematics, the standard form of a circle's equation helps easily identify and visualize it.
A typical equation of a circle in its standard form is expressed as:

• \((x - h)^2 + (y - k)^2 = r^2\)

Here:

• \((h, k)\) represents the circle's center coordinates
• \(r\) is the radius, the distance from the center to any point on the circle

Converting a general conic equation into the standard circle form can reveal whether the conic is indeed a circle. Simply put, if the coefficients \(A\) and \(C\) (of \(x^2\) and \(y^2\)), are equal and \(B = 0\), it simplifies into a circle's equation.

Discriminant in Conics

The discriminant is a crucial tool in identifying the type of conic section represented by a quadratic equation. The general quadratic form is:

• \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\)

The discriminant equation used to classify conic sections is expressed as:

• \(B^2 - 4AC\)

This value helps determine:

• If \(B^2 - 4AC > 0\), the conic section is a hyperbola.
• If \(B^2 - 4AC = 0\), it's a parabola.
• If \(B^2 - 4AC < 0\), it's an ellipse or potentially a circle.

Understanding the discriminant provides a method to categorize conics powerfully, without graphing, by analyzing equations algebraically.

Identifying Conic Sections

Identifying conic sections involves recognizing the specific shape represented by a given equation. These shapes include circles, ellipses, parabolas, and hyperbolas. The initial step is observing the key coefficients \(A\), \(B\), and \(C\) from the general quadratic form. Here's how you can identify conics easily:

• Check the discriminant \(B^2 - 4AC\) to narrow down possibilities (ellipse/circle vs. parabola vs. hyperbola).
• Analyze whether \(A\) equals \(C\), with \(B = 0\); in this case, it indicates a circle.
• Adjustments or completion of squares might sometimes be necessary to simplify and clarify an equation's form.

By systematically applying these methods, one can effectively determine the type of conic section, aiding in understanding shapes and their properties within algebraic boundaries.