Problem 11 - Thomas Calculus

Question

Use the discriminant \(B^{2}-4 A C\) to decide whether the equations represent parabolas, ellipses, or hyperbolas. \(3 x^{2}-5 x y+2 y^{2}-7 x-14 y=-1\)

Step-by-Step Solution

Answer

The equation represents a hyperbola.

Step 1: Identify Coefficients and Rearrange the Equation

The given general quadratic equation is \(3x^{2} - 5xy + 2y^{2} - 7x - 14y = -1\). First, rewrite it in the standard conic section form:\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]Identify the coefficients:- \( A = 3 \)- \( B = -5 \)- \( C = 2 \)- \( D = -7 \)- \( E = -14 \)- \( F = +1 \)

Step 2: Compute the Discriminant

Calculate the discriminant using the formula \( B^2 - 4AC \):\[ B^2 - 4AC = (-5)^2 - 4(3)(2) \]Compute:\[ B^2 = 25 \]\[ 4AC = 4 \times 3 \times 2 = 24 \]\[ B^2 - 4AC = 25 - 24 = 1 \]

Step 3: Interpret the Discriminant

The discriminant value is \(1\), which is greater than zero. Use the following rules to determine the type of conic:- If \( B^2 - 4AC < 0 \), it is an ellipse (if \( A eq C \)) or a circle (if \( A = C \)).- If \( B^2 - 4AC = 0 \), it is a parabola.- If \( B^2 - 4AC > 0 \), it is a hyperbola.Since \(1 > 0\), the equation represents a **hyperbola**.

Key Concepts

Understanding Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. These shapes are fundamental in mathematics and appear frequently in algebra and geometry. The four basic types of conic sections are parabolas, ellipses, hyperbolas, and circles (a special case of ellipses). Each of these shapes has distinct properties and equations that define them.

Conic sections can be represented mathematically in a general quadratic form:

• For circles and ellipses, the defining feature is their round, closed shape.
• Parabolas extend infinitely in one direction, resembling a U-shape.
• Hyperbolas are characterized by their open, outward curves that mirror each other.

By transforming the conic equation into standard form, and calculating the discriminant, we gain insights into the nature of the conic section in question.

Hyperbola: Open and Outward Curving

A hyperbola consists of two disconnected curves, commonly referred to as branches. This shape is formed by intersecting a plane with a cone in such a manner that the plane cuts through both nappes of the cone.

• The primary feature of a hyperbola is that its branches open either vertically or horizontally and are mirror images of each other.
• The general equation of a hyperbola in its standard form is: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).
• The discriminant \( B^2 - 4AC > 0 \) distinguishes hyperbolas from other conic sections.

The given equation in the original problem is determined to be a hyperbola because its discriminant is positive, confirming the presence of this fascinating curve. Hyperbolas have applications in diverse fields such as physics and engineering, particularly in systems that exhibit certain types of symmetry and asymptotic behavior.

Parabolas: The U-Shape Curve

Parabolas are unique among conic sections because they have only one branch. This curve is characterized by a symmetrical U-shape and can open upward, downward, left, or right, depending on its equation.

• A key distinguishing feature of a parabola is its single direction of opening.
• The standard form of a parabola is: \( y = ax^2 + bx + c \) for a vertical opening, or \( x = ay^2 + by + c \) for a horizontal opening.
• In the context of discriminants, \( B^2 - 4AC = 0 \) indicates a parabola.

Parabolas have various real-world applications, especially in projectile motion and satellite dish designs. Their reflective properties make them ideal for focusing light and sound.

Ellipses: The Oval Shape

Ellipses represent closed, oval shapes which may appear circular depending on the equal lengths of their axes. An ellipse is formed when the intersection of the plane is tilted relative to the base of the cone.

• The general equation for an ellipse is: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) when the ellipse is centered at the origin.
• If \( A eq C \) and \( B^2 - 4AC < 0 \), the conic section depicted is an ellipse.
• If \( A = C \), the figure is a circle, which is a special case of an ellipse.

Ellipses have important applications in astronomy, as the orbits of planets and moons are often elliptical in shape. The study of ellipses reveals much about gravitational forces and celestial mechanics.