Problem 30 - Precalculus with Calculus Previews

Question

In Problems \(29-32\), use the discriminant to identify the conic. Rewrite the equation in the form given in (13) and find two functions defined implicitly by this equation. Give the domain of each function. Finally, use a graphing utility to graph these functions on the same rectangular coordinate system. $$ x^{2}-4 x y-2 y^{2}=6 $$

Step-by-Step Solution

Answer

The equation represents a hyperbola and graphs in implicit form.

Step 1: Identify Coefficients

To find the discriminant, identify the coefficients from the given equation \(x^2 - 4xy - 2y^2 = 6\). In standard quadratic form, it is \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Here, \(A = 1\), \(B = -4\), \(C = -2\), \(D = 0\), \(E = 0\), and \(F = -6\).

Step 2: Calculate the Discriminant

The discriminant for conic sections is given by \(B^2 - 4AC\). Substitute the values \(A = 1\), \(B = -4\), and \(C = -2\) into the formula to get \((-4)^2 - 4(1)(-2) = 16 + 8 = 24\).

Step 3: Classify the Conic

Based on the discriminant, classify the conic. If the discriminant is positive, the conic is a hyperbola (\(B^2 - 4AC > 0\)). Since \(24 > 0\), the given equation represents a hyperbola.

Step 4: Rewrite in Standard Form

To rewrite the given equation, factor or complete the square if applicable. Here, it becomes more straightforward by making it equal 0 that is \(x^2 - 4xy - 2y^2 - 6 = 0\) as \(x^2 - 4xy - 2y^2 = 6\) is already derived form.

Step 5: Solve for Functions Implicitly

To find two functions defined implicitly by the equation, solve for \(y\) in terms of \(x\) or vice versa using techniques like solving the quadratic formula if applicable. However, for this conic, it doesn't straightforwardly solve to a form like \(y = f(x)\), indicating its complexity as a hyperbola.

Step 6: Determine the Domain

The domain of each implicit function linked with a hyperbola generally involves all real numbers being applicable where the expression remains real under roots or fractions. The temporal setting here implies graphing software confirms all \(x\) or \(y\) are real within truth.

Key Concepts

Conic Sections

Conic sections are the curves that can be formed by intersecting a plane with a double-napped cone. Each type of conic section has distinct equations and properties. Here are the three most common types:

• Ellipse: A closed, oval-shaped curve. If the intersection of the plane with the cone is perpendicular to the axis of the cone, you get a circle, which is a special type of ellipse.
• Parabola: Formed when the plane is parallel to the slope of the cone. It looks like a 'U' shape, opening in one direction.
• Hyperbola: Created when the plane intersects both nappes of the cone, forming two mirroring curves.

The discriminant, given by the expression \(B^2 - 4AC\), helps determine which conic section we are dealing with when we have a general quadratic equation in two variables. A positive discriminant indicates that the conic section is a hyperbola.

Hyperbola

A hyperbola consists of two disconnected curves known as branches. It is defined by its characteristic equation in the general form: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). For hyperbolas:

• The discriminant \(B^2 - 4AC\) is greater than zero ( > 0).
• It typically represents two symmetric branches that open in opposite directions.
• The standard form can often be represented as \((x-h)^2/a^2 - (y-k)^2/b^2 = 1\), with \(a\) and \(b\) being real numbers that determine the hyperbola's shape and position relative to its center \((h, k)\).

Hyperbolas appear in natural phenomena and technologies:

• In navigation systems like GPS.
• In astronomy when talking about hyperbolic trajectories.

Identifying and understanding a hyperbola's equation enables us to know more about its graphical properties and applications.

Implicit Functions

An implicit function is a relationship between two or more variables expressed through an equation, where one variable is not isolated on one side of the equation. For example, the equation \(x^2 - 4xy - 2y^2 = 6\) describes a hyperbola and is implicit because it does not solve directly to \(y = f(x)\) or \(x = g(y)\).

• Often, implicit functions require techniques from calculus, like implicit differentiation, to find other related expressions or derivatives.
• Graphing tools or algebraic manipulation can be used to analyze or visualize these functions without being explicitly solved for one variable.

When dealing with implicit functions, determining the domain can be more intricate than in simpler `y=` or `x=` functions. For hyperbolas, check for where the expression maintains real and valid values, contributing to understanding its mathematical behavior.