Problem 82 - Thomas Calculus

Question

Suppose you set \(z=0\) in the equation $$ \begin{array}{c}{A x^{2}+B y^{2}+C z^{2}+D x y+E y z+} \\ {F x z+G x+H y+J z+K=0}\end{array} $$ to obtain a curve in the \(x y\) -plane. What will the curve be like? Give reasons for your answer.

Step-by-Step Solution

Answer

Without specific values, the curve is a conic section, such as an ellipse, parabola, or hyperbola.

Step 1: Substitute z = 0 into the equation

Substitute \(z = 0\) into the given equation: \[A x^2 + B y^2 + C \cdot 0^2 + D xy + E y \cdot 0 + F x \cdot 0 + G x + H y + J \cdot 0 + K = 0\] After substitution, this simplifies to:\[A x^2 + B y^2 + D xy + G x + H y + K = 0\] This is the new equation after setting \(z = 0\).

Step 2: Identify the type of curve in the xy-plane

Analyze the equation \[A x^2 + B y^2 + D xy + G x + H y + K = 0\]This equation is a general second-degree equation in \(x\) and \(y\). It can represent a conic section in the \(xy\)-plane such as an ellipse, parabola, or hyperbola, depending on the values of \(A\), \(B\), and \(D\), corresponding to the discriminant \(D^2 - 4AB\).

Step 3: Determine the nature of the conic

Evaluate the expression \(D^2 - 4AB\) to classify the conic section:- If \(D^2 - 4AB < 0\), it is an ellipse (or a circle if \(A = B\) and \(D = 0\)).- If \(D^2 - 4AB = 0\), it is a parabola.- If \(D^2 - 4AB > 0\), it is a hyperbola.Without values for \(A\), \(B\), and \(D\), we can only specify possibilities.

Key Concepts

Ellipse

In geometry, an ellipse is a type of conic section that appears as an elongated circle. It is defined mathematically when the discriminant of a quadratic equation, specifically in the form given by the equation involving variables in the xy-plane, satisfies the condition that

• \(D^2 - 4AB < 0\)
• results in an ellipse.

Think of an ellipse as a squished circle where each point on its circumference is equidistant from two fixed points known as foci. The overall shape depends on the relative values of A and B.
When A equals B, and D is zero, the ellipse becomes a perfect circle.
This symmetry shows in the form of a curve that wraps around these two focal points.

• Key characteristics of an ellipse involve its major and minor axes, which are the longest and shortest diameters of the ellipse respectively.
• Another property is its eccentricity, which quantifies how much the ellipse deviates from being circular.

Ellipses are not only a fascinating mathematical concept but also appear naturally in physics, such as the orbits of planets and satellites.

Parabola

A parabola is another form of conic section that looks like a U-shaped curve. It holds a special significance in both mathematics and the real world. The equation for a parabola in the xy-plane occurs when the discriminant meets the condition:

• \(D^2 - 4AB = 0\)

This indicates a perfect balance between the coefficients of the quadratic terms, leading to a shape defined by symmetry around a central line called the axis of symmetry.
Parabolas can open upwards, downwards, sideways, or in any direction depending on the orientation of the quadratic equation.

• Each parabola has a vertex, which is the highest or lowest point, depending on the direction it's opening.
• The focus and directrix are also important, with the property that any point on the parabola is equidistant from the focus and the directrix.

Parabolas appear in many practical environments, including the path of projectiles or the shape of satellite dishes, showcasing their utility beyond mere curves.

Hyperbola

The hyperbola is a conic section that looks like two mirrored curves opening in opposite directions. This shape emerges under the equation when the quadratic discriminant takes on the condition:

• \(D^2 - 4AB > 0\)

This indicates that the balance of A, B, and D terms results in two branches, which may appear symmetrical yet facing outwardly.
An easy way to define a hyperbola is through two hyperbolic branches that open either up-down or left-right, each approaching asymptotic lines yet never touching them.

• Just like ellipses, hyperbolas have a set of foci, with the distinctive property that the difference in distances from these foci to any point on the hyperbola remains constant.
• The key terms like transverse axis (the line that joins the vertices of the hyperbola) and the conjugate axis (perpendicular to the transverse axis) help in understanding its structure.

Examples of hyperbolas in the real world include the paths of certain orbits and systems under gravitational forces, or the pattern of light beams when they diverge. This elegant conic shape embodies concepts integral to both mathematics and natural phenomena.