Problem 57 - College Algebra with Corequisite Support

Question

Given \(2 x^{2}+k x y+12 y^{2}+10 x-16 y+28=0\) find \(k\) for the graph to be an ellipse.

Step-by-Step Solution

Answer

For the graph to be an ellipse, \(-9.8 < k < 9.8\).

Step 1: Identify the General Conic Equation

The given equation is in the form of a general second-degree polynomial: \[2x^2 + kxy + 12y^2 + 10x - 16y + 28 = 0.\] This equation describes a conic section, which might be an ellipse, hyperbola, parabola, or degenerate case, depending on specific conditions met by the coefficients.

Step 2: Determine Conditions for an Ellipse

The standard form for an ellipse in a conic section is when there is a condition on the discriminant Δ: \[Δ = B^2 - 4AC < 0,\] where A, B, and C are coefficients from the general equation. For the given equation, identify:- \(A = 2\)- \(B = k\)- \(C = 12\).

Step 3: Substitute into the Discriminant Condition

Using the condition for an ellipse, substitute the values we determined:\[Δ = k^2 - 4 \cdot 2 \cdot 12 < 0.\]This simplifies the inequality to: \[k^2 - 96 < 0.\]

Step 4: Solve the Inequality for k

The inequality \(k^2 - 96 < 0\) can be rewritten as \[k^2 < 96.\]Solving this gives the range for \(k\):\(-\sqrt{96} < k < \sqrt{96}.\)Further simplifying \(\sqrt{96}\) gives approximately 9.8, hence\(-9.8 < k < 9.8.\)

Key Concepts

Discriminant Condition

To determine the nature of a conic section, we often use the discriminant condition. The discriminant, denoted as \( \Delta \), helps differentiate between the different types of conic sections like ellipses, hyperbolas, and parabolas. It's derived from the general equation of a conic and expressed as:

• \( \Delta = B^2 - 4AC \)

For an ellipse to form from a conic section, it's vital that this discriminant is less than zero, so \( \Delta < 0 \). This condition ensures that the plot forms the closed symmetric shape distinguishing ellipses from other conic sections. In our example, where \( A = 2 \), \( B = k \), and \( C = 12 \), substituting in these values gives the inequality:

• \( k^2 - 96 < 0 \)

Solving this helps in identifying the range of values for \( k \) that ensure the equation models an ellipse.

General Conic Equation

A general conic equation is a polynomial of degree two that represents various curves like circles, ellipses, parabolas, and hyperbolas. It is typically expressed in the form:

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

This form covers all possibilities of conic sections, based on the coefficients \( A \), \( B \), and \( C \). By shifting these values, the conic formation changes. Specifically for ellipses, additional conditions exist, such as ensuring the discriminant \( \Delta \) is less than zero as mentioned before.
In our given equation \( 2x^2 + kxy + 12y^2 + 10x - 16y + 28 = 0 \), the coefficients indicate a potential ellipse. However, ensuring this demands meeting specific conditions like the discriminant rule highlighted earlier.

Second-degree Polynomial

The basis for understanding and analyzing conic sections lies within second-degree polynomials. These polynomials, also recognized as quadratic equations in two variables, have the highest degree of terms as two. Such equations are fundamental in algebra and are represented as:

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

Each conic shape results from varied combinations of the coefficients \( A \), \( B \), and \( C \), influencing the graph's properties and whether it turns into an ellipse or another form. Delving into our given mathematical problem, it serves as a quintessential application of second-degree polynomials to explore real-world shapes and analyze their properties mathematically, opening the avenue to understanding ellipses through the coefficient interaction.