Problem 11 - College Algebra

Question

For the following exercises, determine which conic section is represented based on the given equation. $$2 x^{2}+3 y^{2}-8 x-12 y+2=0$$

Step-by-Step Solution

Answer

The equation represents an ellipse.

Step 1: Identify the General Form

The given equation is in the form \( ax^2 + by^2 + cx + dy + e = 0 \), which is a general form for a conic section. For the provided equation, \( a = 2 \), \( b = 3 \), \( c = -8 \), \( d = -12 \), and \( e = 2 \).

Step 2: Determine Values of a and b

For the given quadratic equation, observe the coefficients of \( x^2 \) and \( y^2 \): \( a = 2 \) and \( b = 3 \).

Step 3: Compare Values of a and b

To classify the type of conic section, compare the coefficients \( a \) and \( b \): - If \( a = b \), it is a circle. - If \( a eq b \) and both are positive, it is an ellipse. - If \( a \cdot b < 0 \), it is a hyperbola. - If either \( a = 0 \) or \( b = 0 \), it is a parabola.For the given equation, \( a = 2 \) and \( b = 3 \) are positive and not equal.

Step 4: Conclude the Type of Conic

Since \( a = 2 \) and \( b = 3 \) are both positive and not equal, it can be concluded that the equation represents an ellipse.

Key Concepts

Ellipse

An ellipse is a type of conic section that you can recognize by its elongated circle shape. But in mathematical terms, it’s not just about the shape. An ellipse has a unique property where the sum of the distances from any point on its curve to two fixed points, called foci, remains constant. This definition gives ellipses their oval form.
When dealing with equations, an ellipse becomes apparent if the coefficients of the squared terms are both positive and not equal.

• If the coefficients were equal, it would form a perfect circle, a special case of the ellipse.
• If one of the squared term coefficients was zero, the conic section wouldn't be an ellipse.

While circles can be seen as simple ellipses, most ellipses are stretched along their major and minor axes. Recognizing this by examining equations is vital in mathematics, particularly if you're given a general form conic section equation.

General Form of Conic Sections

Conic sections can take on a variety of shapes, including ellipses, parabolas, and hyperbolas. These shapes can be derived from the general form equation: \[ ax^2 + by^2 + cx + dy + e = 0 \]This equation may look daunting, but it holds the key to unlocking the specific type of conic section you're dealing with. Each component of this equation plays a role in determining the conic's characteristics.

• The coefficients \(a\) and \(b\) in front of \(x^2\) and \(y^2\) are crucial. Their values relative to each other define whether the conic is an ellipse, parabola, circle, or hyperbola.
• The terms \(cx\) and \(dy\) represent linear components that affect the position and orientation of the conic section.
• The constant \(e\) influences the conic’s placement on the coordinate plane but doesn't define its type.

By organizing your given equation into this general form, you can easily compare coefficients and identify the type of conic section you're working with.

Coefficients Comparison in Conics

Coefficients play a fundamental role in identifying and classifying conic sections. When dissecting an equation in the general conic form, paying specific attention to the coefficients \(a\) and \(b\) helps determine which conic section is represented.

• If \(a = b\), the conic section is a circle.
• If \(a \, \text{and} \, b\) are both positive but not equal, the conic is an ellipse, like in our provided equation where \(a = 2\) and \(b = 3\).
• If \(a \, \cdot \, b < 0\), the conic is a hyperbola as the shape will open either horizontally or vertically.
• If either \(a = 0\)\(\text{ or } b = 0\), the conic is a parabola, meaning the graph represents a curve with one continuous arc.

Understanding these relationships between the coefficients allows you to quickly classify the conic section, making this step a key skill for students.