Problem 307 - Calculus Volume 3

Question

Determine which of the conic sections is described. $$ x^{2}+4 x y-2 y^{2}-6=0 $$

Step-by-Step Solution

Answer

The conic section is a hyperbola.

Step 1: Identifying the General Conic Equation

The given equation is \( x^2 + 4xy - 2y^2 - 6 = 0 \). This can be compared to the general conic section equation: \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). By comparing, we identify: \( A = 1 \), \( B = 4 \), \( C = -2 \), \( D = 0 \), \( E = 0 \), \( F = -6 \).

Step 2: Calculating the Discriminant

The discriminant \( \, ext{of a conic is given by} \, D = B^2 - 4AC \.\) For our equation: \( B = 4 \), \( A = 1 \), and \( C = -2 \). Therefore, the discriminant \( D = 4^2 - 4(1)(-2) = 16 + 8 = 24 \).

Step 3: Determining the Type of Conic Section

To determine the type of conic, we use the discriminant: if \( D > 0 \), it's a hyperbola; if \( D = 0 \), it's a parabola; if \( D < 0 \), it's an ellipse or a circle (if \( A = C \)). Since \( D = 24 \), which is greater than 0, the conic is a hyperbola.

Key Concepts

Hyperbola

In mathematics, a hyperbola is one of the four types of conic sections. Conic sections are shapes created by the intersection of a plane with a cone. Just like ellipses and parabolas, hyperbolas are part of this family of curves, each shaped by different plane and cone intersections. A hyperbola consists of two separate curves called branches, which resemble mirrored arches. These branches open away from each other. Hyperbolas have two axes of symmetry, a transverse axis and a conjugate axis.

One way to identify a hyperbola is through its equation. A hyperbolic equation in its simplest form may look like this: \[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]or:\[\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1\]where \((h, k)\) is the center of the hyperbola, and \(a\), \(b\) control the shape and orientation of the graph. It can be horizontally oriented (the first equation) or vertically oriented (the second equation). Hyperbolas are commonly found in various real-world applications, such as radio transmission, navigation systems, and astronomy.

Conic Discriminant

The conic discriminant is a crucial tool for determining the type of conic section represented by a given quadratic equation. The general form of such an equation is:\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\]Where \(A\), \(B\), and \(C\) are coefficients of the quadratic terms, and \(D\), \(E\), \(F\) are the coefficients of the linear terms and constant, respectively.

Use the discriminant \( D \) to identify the type of conic. The discriminant for a conic section is calculated by:\[D = B^2 - 4AC\]

• If \( D > 0 \), the equation represents a hyperbola.
• If \( D = 0 \), it is a parabola.
• If \( D < 0 \), the equation is an ellipse. If additionally, \( A = C \), the ellipse is a circle.

In our example, the discriminant \( D = 24 \), which is greater than 0, reveals that the conic section described is a hyperbola. Understanding and calculating the conic discriminant is a reliable method for quickly identifying the type of curve represented by an equation.

General Conic Equation

Every conic section can be described using a general conic equation, a fundamental part of algebra and geometry. This equation takes the form:\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\]Here, each term corresponds to the various parts of a curve. Values of the coefficients \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\):

• \(Ax^2\) and \(Cy^2\) show how much \(x^2\) and \(y^2\) contribute to the curve.
• \(Bxy\) defines the inclination or orientation of the curve.
• \(Dx\), \(Ey\), and \(F\) can shift, tilt, or move the entire curve without changing its basic shape.

Each conic section type—circle, ellipse, parabola, and hyperbola—can be identified by looking at these coefficients. The intersection of the coefficients determines the conic's shape and position. By analyzing these elements, particularly through the discriminant \( D \), you can distinguish whether the conic section is a hyperbola, as in our specific example, or another shape. Understanding this equation thoroughly helps in graphing and analyzing conic sections efficiently.