Problem 21 - A Graphical Approach to Precalculus with Limits

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. $$2 x^{2}-8 x+2 y^{2}+20 y=12$$

Step-by-Step Solution

Answer

The equation represents a circle.

Step 1: Write the Equation in Standard Form

Start by rearranging the given equation so that all terms are on one side and equal to zero. We want the equation in the form of \[ Ax^2 + By^2 + Cx + Dy + E = 0 \]Given equation is: \[ 2x^2 - 8x + 2y^2 + 20y = 12 \]Subtract 12 from both sides to set the equation to zero: \[ 2x^2 - 8x + 2y^2 + 20y - 12 = 0 \]

Step 2: Identify the Coefficients

Identify the coefficients of the terms in the equation for comparison to the conic section forms: \[ A = 2, \ B = 2, \ C = -8, \ D = 20, \ E = -12 \] Here, \( A = 2 \) and \( B = 2 \) are the coefficients for \( x^2 \) and \( y^2 \) respectively, both of which are equal.

Step 3: Determine the Type of Conic Section

According to the standard forms of conics: - **Circle**: \( A = B \), \( C = D = 0 \)- **Ellipse**: \( A eq B \)- **Parabola**: Only one squared term (either \( A \) or \( B \) is zero)- **Hyperbola**: \( A \) and \( B \) have opposite signs.Here both \( A \) and \( B \) are equal (\( 2 = 2 \)) and positive, and although \( C \), \( D \), and \( E \) are non-zero, the equality of \( A \) and \( B \) suggests it's a circle.

Key Concepts

Circle

A circle is a basic shape in geometry that is perfectly round. It is defined as the set of all points in a plane that are at a constant distance, called the radius, from a fixed point, known as the center.
In terms of equations, a circle is a special kind of conic section. Unlike other conic sections, a circle forms when a plane cuts through a cone parallel to its base. When considering the equation form, if the coefficients of the squared terms in the equation of a conic section are equal, the shape is a circle.
For example, in the equation 2x² + 2y², since the coefficients (2 for x² and 2 for y²) are equal, this suggests the graph of this equation is a circle.

Equation of Conic Sections

Conic sections describe the curves obtained by intersecting a cone with a plane at different angles. The primary types of conic sections include circles, ellipses, parabolas, and hyperbolas, each having a distinct shape and equation.
For any conic section, the general form of an equation is given by: \[ Ax^2 + By^2 + Cx + Dy + E = 0 \]This form helps determine the type of conic based on its coefficients. The coefficients A and B, especially when looking at their equality and signs, help define the type of conic section:

• **Circle:** When A equals B and both are positive.
• **Ellipse:** When A is not equal to B, and both are positive.
• **Parabola:** When either A or B is zero, meaning only one variable is squared.
• **Hyperbola:** When A and B have opposite signs.

By understanding the relationship between these coefficients, we can identify the conic section without plotting the graph.

Standard Form of Conic Sections

The standard form of conic sections allows us to write equations in a way that their geometric properties are immediately apparent. This format simplifies identifying the type of conic section and its features, such as center, vertex, or axes.
A conic section in standard form is rearranged so that it has all terms on one side of the equation, making it easier to identify its type:

• **Circle:** \( x^2 + y^2 = r^2 \), where r is the radius and the center is at the origin.
• **Ellipse:** \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \((h, k)\) is the center.
• **Parabola:** \( y = ax^2 + bx + c \) or \( x = a(y-k)^2 + h \), representing a simple quadratic equation.
• **Hyperbola:** \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \), showing two symmetric curves.

Correctly rearranging a given equation to this standard form not only helps in identifying the type of conic section but also aids in visualizing it geometrically.