Problem 25 - Precalculus with Calculus Previews

Question

In Problems \(23-28,\) use the discriminant to identify the conic without actually graphing. $$ 4 x^{2}-4 x y+y^{2}-6=0 $$

Step-by-Step Solution

Answer

This equation represents a parabola.

Step 1: Identify the equation form

The equation given is a general second-degree equation in two variables: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). By comparing, we identify coefficients: \(A=4\), \(B=-4\), \(C=1\), \(D=0\), \(E=0\), and \(F=-6\).

Step 2: Calculate the discriminant

The discriminant \(\Delta\) for a conic section is calculated using the formula \(\Delta = B^2 - 4AC\). Substitute the values: \(\Delta = (-4)^2 - 4 \cdot 4 \cdot 1 = 16 - 16 = 0\).

Step 3: Interpret the discriminant

The value of the discriminant \(\Delta = 0\) indicates that the conic section is a parabola, as per the rules for discriminant interpretation for conics: \(\Delta > 0\) for hyperbolas, \(\Delta = 0\) for parabolas, and \(\Delta < 0\) for ellipses, which includes circles if \(A = C\).

Key Concepts

Discriminant

The discriminant is a key mathematical concept that helps determine the nature of conic sections. For any conic section represented by a second-degree equation of the form: \\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]The discriminant \( \Delta \) is calculated using the formula:

• \( \Delta = B^2 - 4AC \)

This simple expression gives insight into what type of conic section the equation represents:

• If \( \Delta > 0 \), the conic is a hyperbola.
• If \( \Delta = 0 \), the conic is a parabola.
• If \( \Delta < 0 \), the conic is an ellipse. A circle is a special form of an ellipse where \( A = C \).

Make sure to calculate the discriminant accurately. It splits the plane into clearly defined shapes without requiring the graphing of the equation.

Parabolas

Parabolas are one of the fundamental shapes in the family of conic sections. A critical identification feature of a parabola is when the discriminant, \( \Delta \), equals zero. This distinguishes it from other conics:
Parabolas have several characteristics:

• They are symmetrical, with a line of symmetry running through their vertex.
• They open either vertically or horizontally, depending on their orientation.
• The vertex is the point where the parabola turns.

In practical applications, parabolas appear in diverse areas such as:

• Projectile motions, where the trajectory is parabolic.
• Reflector technologies, like satellite dishes and headlights.

Understanding this shape from its equation without plotting simplifies the analysis and prediction of a parabola's behavior.

Second-Degree Equations

Second-degree equations, or quadratic equations, are pivotal in describing conic sections. These equations take the form:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]Where:

• \( A, B, C \) are coefficients that determine the equation's specific type.
• \( D, E \) are coefficients for linear terms in \( x \) and \( y \).
• \( F \) is a constant term.

The classification of these equations leads to distinct shapes like parabolas, hyperbolas, ellipses, and circles. By understanding the relationship between the coefficients and their discriminant \( \Delta \), we can discern the particular type of conic section at play. Exploring these second-degree equations helps in not only identifying the shape but also in understanding its properties, opening a deeper insight into analytic geometry.