Problem 26 - Precalculus with Calculus Previews

Question

In Problems \(23-28,\) use the discriminant to identify the conic without actually graphing. $$ x^{2}+\sqrt{3} x y-\frac{1}{2} y^{2}=0 $$

Step-by-Step Solution

Answer

The conic is a hyperbola.

Step 1: Recall the General Form of a Conic Section

The general form for conic sections is given by the equation: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \] Identify the coefficients: - \( A = 1 \) - \( B = \sqrt{3} \) - \( C = -\frac{1}{2} \)

Step 2: Determine the Discriminant

The discriminant \( \Delta \) for a conic section is calculated as \[ \Delta = B^2 - 4AC \] Substitute the values we found:\[ \Delta = (\sqrt{3})^2 - 4 \cdot 1 \cdot \left(-\frac{1}{2}\right) \] \[ \Delta = 3 + 2 = 5 \]

Step 3: Interpret the Discriminant

The value of the discriminant \( \Delta \) determines the type of conic section:- \( \Delta < 0 \): ellipse- \( \Delta = 0 \): parabola- \( \Delta > 0 \): hyperbolaSince \( \Delta = 5 > 0 \), the conic is a hyperbola.

Key Concepts

Discriminant of Conic Sections

In the study of conic sections, the discriminant plays a crucial role in determining the nature of the conic without having to graph it. For an equation to be considered in the form of a conic, it should be expressed as: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \] where \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\) are constants. A vital part of this equation is the discriminant \( \Delta \), which is determined using the following formula: \[ \Delta = B^2 - 4AC \] The discriminant offers insights into the conic's geometric nature based on the relationships among the coefficients:

• \( \Delta < 0 \): Indicative of an ellipse, suggesting that the conic may take a circular or elliptical shape based on \(A\) and \(C\).
• \( \Delta = 0 \): Represents a parabola, characterized by its unique U-shape, either opening along the x-axis or y-axis.
• \( \Delta > 0 \): Identifies a hyperbola, which consists of two disconnected curves mirroring one another.

Understanding this concept allows mathematicians and students to quickly ascertain the conic type, speeding up analysis and problem-solving when graphing isn’t feasible.

Types of Conics

Conics represent a fascinating group of curves generated by intersecting a plane with a double-napped cone. When classified based on their discriminant, they help reveal the nature of the intersection:

• Circle: A special case of an ellipse where the plane cuts perpendicular to the cone's axis, yielding a perfectly symmetric shape.
• Ellipse: Formed when a tilted plane intersects the cone at an angle, not parallel to the base, resulting in an oval shape. An ellipse occurs when \( \Delta < 0 \).
• Parabola: Occurs when the intersecting plane is parallel to the slant edge of the cone, creating a distinct U-shaped curve, with characteristics determined by \( \Delta = 0 \).
• Hyperbola: Takes form when the plane cuts both nappes of the cone, producing two symmetric, open curves. Identified by \( \Delta > 0 \), as in our example problem.

These types illustrate the diversity and interplay of conic sections and their crucial role in fields ranging from physics to engineering, where they model planetary orbits or reflect light rays.

Hyperbola Identification

Identifying a hyperbola among conic sections can be achieved quickly by understanding its unique properties. Comprising two mirrored curves, a hyperbola is discovered using the discriminant: \(\Delta = B^2 - 4AC > 0\) An important feature of a hyperbola is that it's represented by equations where both quadratic terms \(Ax^2\) and \(Cy^2\) have opposing signs. This results in the conic intersecting the plane at an angle that exposes both cones. Understanding the geometry:

• A hyperbola can appear horizontally or vertically, contingent upon the signs and magnitudes of \(A\) and \(C\).
• Its standard equations vary:
  • Horizontal hyperbola: Difference between squares: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
  • Vertical hyperbola: Reversed equation: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\)
• The foci are located outside its vertices, and the hyperbola opens around these foci, emphasizing the space between its branches.

Hyperbolas are key in depicting scenarios containing two distant points of view, such as in radio wave transmission and satellite engagements.