Problem 12 - College Algebra with Corequisite Support

Question

For the following exercises, determine which conic section is represented based on the given equation. \(4 x^{2}+9 x y+4 y^{2}-36 y-125=0\)

Step-by-Step Solution

Answer

The given equation represents a hyperbola.

Step 1: Identify the Equation Components

The given equation is \(4x^2 + 9xy + 4y^2 - 36y - 125 = 0\). Here, we observe the presence of both \(x^2\) and \(y^2\) terms, a mixed \(xy\) term, and constant terms. Our task is to identify which type of conic section this equation represents by analyzing these components.

Step 2: Analyze the Coefficients of Quadratic Terms

The equation can be compared to the general conic section form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). In the given equation, \(A = 4\), \(B = 9\), and \(C = 4\). Since the \(xy\) term is present, this may indicate rotation, and we need to calculate \(B^2 - 4AC\) to determine the conic type.

Step 3: Calculate the Discriminant

Calculate the value of the discriminant \(B^2 - 4AC\). Here, \(B = 9\), \(A = 4\), and \(C = 4\). Substitute these values into the discriminant formula: \(B^2 - 4AC = (9)^2 - 4(4)(4)\).

Step 4: Compute the Discriminant Value

Perform the calculation: \(B^2 = 81\) and \(4AC = 64\), thus \(B^2 - 4AC = 81 - 64 = 17\).

Step 5: Determine the Conic Section Type

For conic sections, if \(B^2 - 4AC > 0\), the conic is a hyperbola. Here, \(B^2 - 4AC = 17 > 0\), which indicates that the given equation represents a hyperbola.

Key Concepts

Conic Section Identification

Conic sections are fascinating curves that appear in many physical and mathematical applications. These curves are the intersections of a plane with a double napped cone, generating different shapes based on the angle and location of the intersection. The four main types of conic sections are circles, ellipses, parabolas, and hyperbolas. These can be identified using a quadratic equation of the form: \\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]Understanding conic sections requires looking at the different components present in the equation.\

• **Circle**: If the equation is in the form \(Ax^2 + Ay^2 + Dx + Ey + F = 0\) and \(A = C\) with no \(xy\) term, it's a circle.
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• **Ellipse**: Similar to the circle, but here \(A eq C\), still with no \(xy\) term.
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• **Parabola**: Either \(B = 0\) and one of \(A\) or \(C\) is zero, or the \(xy\) is present but doesn't change the type.
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• **Hyperbola**: There's an \(xy\) term or \(B^2 - 4AC > 0\).

Recognizing these terms and configurations helps in determining which conic section the given equation represents.

Discriminant of a Conic Section

The discriminant plays a vital role in conic section identification. It provides a quick way to identify the type of conic section represented by a quadratic equation. The discriminant formula is: \\[ B^2 - 4AC \] in the equation \\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]The value of the discriminant helps classify the conic section:

• If \(B^2 - 4AC < 0\), the conic section is an ellipse. A special case of this condition is when \(A = C\), resulting in a circle.
• If \(B^2 - 4AC = 0\), the conic is a parabola. This indicates that the plane is tangent to the cone, producing a parabolic path.
• If \(B^2 - 4AC > 0\), the equation represents a hyperbola. This condition reflects two distinct curves ("branches") on the coordinate plane.

By calculating the discriminant, one can efficiently determine the type of conic section without graphing it.

Hyperbola Identification

Hyperbolas are a unique type of conic section that appear when the discriminant \(B^2 - 4AC\) is positive. These shapes are characterized by two separate pieces, known as branches, which mirror each other across a central region.When identifying a hyperbola in the equation form \\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \], the presence of an \(xy\) term typically signifies a potential rotation of the axes. However, the telling sign of a hyperbola is the positive discriminant discussed earlier.Some key features of hyperbolas include:

• They open either horizontally or vertically, depending on the terms involved.
• The vertices, centers, and asymptotes of a hyperbola provide insight into its orientation and position on the plane.

In our exercise, the equation \(4x^2 + 9xy + 4y^2 - 36y - 125 = 0\) has a positive discriminant \(B^2 - 4AC = 17\), confirming it represents a hyperbola. Understanding the properties of hyperbolas allows for deeper exploration into complex systems and applications, such as in physics and engineering.