Problem 13 - College Algebra

Question

For the following exercises, determine which conic section is represented based on the given equation. $$3 x^{2}+6 x y+3 y^{2}-36 y-125=0$$

Step-by-Step Solution

Answer

The equation represents a parabola.

Step 1: Identify the general form of the conic equation

The given equation is \(3x^{2}+6xy+3y^{2}-36y-125=0\). This can be expressed in the standard form of a second-degree polynomial: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), where \(A = 3\), \(B = 6\), \(C = 3\), \(D = 0\), \(E = -36\), and \(F = -125\).

Step 2: Calculate the discriminant (B² - 4AC)

To determine the type of conic section, we calculate the discriminant using the formula \(B^2 - 4AC\). Substitute the values of \(A\), \(B\), and \(C\) into this formula: \(B^2 = 6^2 = 36\) and \(4AC = 4 \times 3 \times 3 = 36\). Thus, the discriminant \(B^2 - 4AC = 36 - 36 = 0\).

Step 3: Interpret the discriminant

The discriminant \(B^2 - 4AC\) helps us identify the type of conic section. If \(B^2 - 4AC = 0\), the conic section is a **parabola**. Thus, the given equation represents a parabola.

Key Concepts

Parabola

In conic sections, a parabola is an open curve that can be represented by a quadratic equation. Parabolas are incredibly important in mathematics and physics, showing up in situations such as projectile motion and satellite dishes. When examining conic sections through algebraic equations, the presence of a parabola is confirmed when the discriminant equals zero.

A parabola has certain features that make it unique:

• Vertex: The point where the parabola changes direction.
• Focus: A fixed point used to generate the curve, located inside the curve.
• Directrix: A line that is perpendicular to the axis of symmetry of the parabola.

Understanding these properties helps in graphing a parabola accurately. Remember, parabolas can open upwards, downwards, or sideways, depending on the orientation of the axis of symmetry.
Developing intuition about parabolas is a great asset in geometry and algebra.

Discriminant

The discriminant is a mathematical expression that assists in determining the type of conic section represented by a given second-degree polynomial equation. Calculated using the formula \( B^2 - 4AC \), the discriminant helps identify whether the graph of the equation aligns with one of several conic shapes.

For conic sections, here's how the discriminant is interpreted:

• If \( B^2 - 4AC > 0 \), the conic section is an ellipse (excluding the circle) or a hyperbola.
• If \( B^2 - 4AC = 0 \), it indicates a parabola.
• If \( B^2 - 4AC < 0 \), it suggests an ellipse (including the circle under specific conditions).

Using the discriminant makes it easier to quickly classify the type of conic section, thus aiding problem-solving in algebraic contexts.

Second-degree Polynomial

A second-degree polynomial, also known as a "quadratic polynomial", features terms up to the second power. In two variables, such as \( x \) and \( y \), the general form is given by:

\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
This notation is fundamental for studying conic sections, as it encompasses circles, ellipses, parabolas, and hyperbolas, depending on the coefficients \( A \), \( B \), and \( C \).

Key characteristics include:

• The coefficient \( A \) influences the shape and orientation of the graph, especially in parabolic sections.
• The presence of the term \( Bxy \) suggests a rotation of axes, which can provide additional geometric insights.
• Understanding how to rearrange and interpret these terms is a fundamental aspect of identifying and graphing specific conic sections.

Grasping the basics of second-degree polynomials is crucial for exploring the rich field of conic sections.