Problem 75 - Precalculus with Limits

Question

TRUE OR FALSE? In Exercises 73-76, determine whether the statement is true or false. Justify your answer. If \(D\neq0\) and \(E\neq0\), then the graph of \(x^2-y^2+Dx+Ey=0\) is a hyperbola.

Step-by-Step Solution

Answer

True, the given equation represents a hyperbola because it follows the general formula of the hyperbola equation.

Step 1: Identify the general form of hyperbola

The standard form of the equation of a Hyperbola is \(Ax^2 + By^2 + Cx + Dy + E = 0\), where \(A != B\) and neither \(A\) or \(B\) can be zero. The signs in front of \(x^2\) and \(y^2\) must be opposite, i.e. one positive and one negative.

Step 2: Compare given equation

The given equation is \(x^2 - y^2 + Dx + Ey = 0\). Here, the coefficients of \(x^2\) and \(y^2\) are 1 and -1, which are not equal and neither of them are zero. The coefficients of \(x\) and \(y\) are \(D\) and \(E\) respectively which are mentioned in the question not to be zero.

Step 3: Conclusion of comparison

Because the given equation follows all the rules and norms of the general form of a hyperbola, it's safe to conclude that it represents a hyperbola.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. There are four main types of conic sections; circles, ellipses, parabolas, and hyperbolas. Each shape is defined by the angle at which the plane intersects the cone. This mathematical concept extends into many real-world applications, such as satellite dishes and planetary orbits.

• **Circle**: A circle is formed when the intersecting plane is perpendicular to the cone's axis.
• **Ellipse**: This occurs when the plane intersects the cone at an angle, resulting in an elongated circle.
• **Parabola**: It forms when the plane is parallel to an element of the cone, creating a U-shaped curve.
• **Hyperbola**: This shape emerges when the plane intersects both nappes of the cone at an angle, resulting in two separate curves.

Understanding conic sections helps in recognizing how different equations govern the types of conics. Each type has a distinct equation form, characterized by specific coefficient conditions and signs.

Hyperbola

A hyperbola is a type of conic section that appears as two distinct, mirrored curves. It can be thought of as two infinite bows or mirrored parabolas placed back-to-back. The defining feature of a hyperbola is its equation, where the degree of the terms is exactly the same but their coefficients have opposite signs.

• The standard equation of a hyperbola is typically expressed as \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\), depending on the orientation.
• The signs of coefficients for the \(x^2\) and \(y^2\) parts are crucial because they must be opposite; otherwise, the equation might represent an ellipse or a different conic section.
• Hyperbolas have two axes of symmetry — the transverse axis and the conjugate axis. These axes help to determine the direction and shape of the hyperbola.

In the context of the given equation \(x^2 - y^2 + Dx + Ey = 0\) from the original exercise, the equation follows the form of a hyperbola since the coefficients of \(x^2\) and \(y^2\) are positive and negative respectively, making it possible to identify the graph as such.

Algebraic Equations

Algebraic equations are mathematical statements that express the equality between two expressions. They are essential in describing and solving for mathematical relationships and are pivotal in defining the characteristics of conic sections.

• An algebraic equation comprises constants, coefficients, variables, and operators organized to define a relationship.
• The structure of the equation often determines what kind of graph it will represent, such as a line, parabola, ellipse, or hyperbola.
• In precalculus, understanding the types and structures of equations is key to solving and graphing these mathematical relationships accurately.

When tackling problems like \(x^2 - y^2 + Dx + Ey = 0\), it is integral to compare the form of the equation to known standard forms. This allows us to discern its graphical representation, such as identifying its form as a hyperbola due to the opposite signs of \(x^2\) and \(y^2\). Understanding algebraic structures enables deeper insights into the properties and solutions of these equations.