## Exploring Analyic Geometry with |
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Equations and Graphs

Variables and Functions

Polynomials

Equations

Solving Equations

Graphs

Parametric Equations

Explorations

Using algebraic techniques to solve geometry problems is the difference in approach between analytic geometry and planar geometry. Use of such techniques links the algebraic concept of an equation to the graphical representation of geometry shown in a graph or plot. This chapter introduces some of the simple algebraic techniques for solving equations that are heavily used in analytic geometry.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Variables and Functions [Top]

A variable is a quantity to which arbitrary values may be assigned. Let x be a symbol representing such a variable and let the quantity represented by the symbol y depend on x. We call y a function of x and say that x is the independent variable, and y the dependent variable. Using standard mathematical notation, these statements are written as y=f(x) and is read "y is a function of x." The value of the function at x=a is written f(a). These definitions may be expanded so that a variable z depends on two independent quantities x and y (as in solid analytic geometry), and relationships of this type are written z=f(x,y).

A function y=f(x) is real-valued if y is real when x is real. If there is but one value of y for a given value of x, y is said to be a single-valued function. If, for a given value of x, y has more than one value, y is said to be multiple-valued. The function f(x) is periodic if f(x+P)≡f(x) for some period, P. Usually it is assumed that P is the least number for which this identity is true.

Polynomials [Top]

A mathematical expression consisting of a sum of various positive integer powers of a variable is called a polynomial. The largest exponent that appears in a polynomial is called the degree of the polynomial. Polynomials of low-degree have special names as shown in Table [eqn:tbl01].

Low-degree polynomials.

Polynomials can involve more than one variable. For example the polynomial x+2y+3 is a linear polynomial in two unknowns and is a quadratic polynomial in two unknowns. Descarta2D provides special objects, called equation objects, for representing linear and quadratic polynomials in two unknowns (see Table [eqn:tbl02]).

Descarta2D equation objects.

Example: Convert the polynomials 4x-2y+1 and into equivalent Line2D and Quadratic2D objects. Perform the inverse conversions.

Solution: Line2D[poly,{x,y}] and Quadratic2D[poly,{x,y}] convert linear and quadratic polynomials into equivalent Line2D and Quadratic2D objects. The functions Polynomial2D[line,{x,y}] and Polynomial2D[quad,{x,y}] convert Line2D and Quadratic2D objects, respectively, into polynomials.

Clear[x,y];

{l1=Line2D[4*x-2*y+1,{x,y}],

q1=Quadratic2D[x^2-3*x*y+3*x-2*y+4,{x,y}]}

{Polynomial2D[l1,{x,y}],Polynomial2D[q1,{x,y}]}

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Equations [Top]

If a function of a single variable, f(x), is set equal to zero, the relation f(x)=0 is called an equation. This equation imposes a condition on the variable x which then can assume only certain values. For example, if Ax+B=0, then x can take on only one value, x=-B/A. If the equation is sinx=0, x can assume an unlimited number of values of the form kπ, where k is any integer. The process of finding the values of x that satisfy the equation is called solving the equation. The values of x which satisfy f(x)=0 are called the solutions or roots of the equation. All of the real solutions of f(x)=0 may be represented by points on a line such as the x-axis. These points constitute the graph of the equation in one dimension.

If a function of two variables, f(x,y), is set equal to zero the relation f(x,y)=0 is also an equation. But this equation permits one of the variables to be independent, while the other is dependent and a function of the first. For example, f(x,y)=0 might be solved for y in terms of x, yielding , indicating that x is the independent variable and y the dependent variable. Or f(x,y)=0 might be solved for x yielding interchanging the independent and dependent variables.

Descarta2D objects, polynomials and equations.

In addition to representing polynomials, the Line2D and Quadratic2D objects may also be used to represent equations (the implicit assumption is that they represent polynomials set equal to zero). Figure [eqn:fig01] shows the relationships between polynomials, equations and Descarta2D equation objects. Table [eqn:tbl03] summarizes the Descarta2D functions that accomplish the conversions labeled 1 to 4 in Figure [eqn:fig01].

Descarta2D conversion functions.

Example: Convert the Descarta2D linear equation object Line2D[2,3,-1] into an equivalent Mathematica equation. Similarly, convert the Descarta2D quadratric object Quadratic2D[1,-2,2,3,-3,7] into a Mathematica equation.

Solution: The Descarta2D function Equation2D[line,{x,y}] converts a Line2D object into a Mathematica equation. The function Equation2D[quad,{x,y}] converts a Quadratic2D object into a Mathematica equation.

Clear[x,y];

{Equation2D[Line2D[2,3,-1],{x,y}],

Equation2D[Quadratic2D[1,-2,2,3,-3,7],{x,y}]}

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Solving Equations [Top]

In our study of analytic geometry we will often need to solve linear and quadratic equations. We will also need to solve systems of two or more equations. Mathematica provides functions for solving individual equations and systems of equations, either exactly (the Solve function) or numerically (the NSolve function). The following subsections illustrate the use of these Mathematica functions.

One Linear, One Unknown

The equation ax+b=0 is a linear equation in one unknown. By simple algebra, the solution to this equation is x=-b/a. The equation is invalid (or trivial) and has no solution if a=0.

Example: Solve the equation 3x+12=0.

Solution: The Mathematica function Solve[eqn,variable] returns a list of solutions for an equation in one unknown. The solution(s) are returned in the form of Mathematica rules.

Clear[x];

Solve[3x+12==0,x]

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One Quadratic, One Unknown

The quadratic equation has two solutions

The expression under the radical, , is called the discriminant of the equation and determines the type of solutions admitted by the equation. Assuming the coefficients are real numbers, D>0 indicates that the equation has two real and distinct solutions; if D=0 the equation has two real solutions that are equal; and if D<0 the equation has two complex solutions that are conjugates of each other.

Example: Find the solutions of the equation .

Solution: The Mathematica function Solve[eqn,variable] returns a list of solutions for an equation in one unknown. The solution(s) are returned in the form of Mathematica rules.

Clear[x];

Solve[3x^2-4x-5==0,x]

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Two Linears, Two Unknowns

A list of two or more equations that are to be solved simultaneously is called a system of equations. Consider the system of two linear equations

Simple algebra yields the formulas for x and y that solve the two equations:

If the denominator, , is equal to zero the equations have no solution and are called inconsistent.

Example: Find the solution of the two linear equations x-3y+4=0 and 2x+5y-3=0.

Solution: The Mathematica function Solve[eqnList,varList] returns a list of solutions for a system of equations in several variables. The solution(s) are returned in the form of Mathematica rules.

Clear[x,y];

Solve[{x-3y+4==0,2x+5y-3==0},{x,y}]

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One Linear, One Quadratic, Two Unknowns

Consider the linear and quadratic equations

. | . | |

. | . |

In the general case the system of these two equations can be solved by first solving the linear equation for one of the variables, say x, in terms of the other, y. The expression for x is then substituted into the quadratic equation, yielding a somewhat more complicated quadratic equation in y alone. The quadratic equation in one variable is then solved yielding two values for y which may then be substituted back into the linear equation to determine the corresponding values of x. While this solution technique is straightforward, it produces somewhat complicated expressions for x and y, and special cases must be handled individually (for example, if the linear equation has no y term, then the procedure must be altered to solve for x instead).

Example: Solve the system of equations

using the Mathematica Solve command.

Solution: The Mathematica function Solve[eqnList,varList] returns a list of solutions for a system of equations in several variables. The solution(s) are returned in the form of Mathematica rules.

Clear[x,y];

ans=Solve[{3x+4y-1==0,2x^2+y^2+6x-4y+1==0},{x,y}]

These somewhat complicated solutions can be approximated by decimal numbers using the Mathematica N function.

N[ans]

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Two Quadratics, Two Unknowns

The system of two quadratic equations in two unknowns

. | and | |

. | . |

can be solved algebraically using a technique involving a pencil of the two quadratic equations. This technique will be discussed in more detail in later chapters. Even though the technique can yield a symbolic formula for the solutions, such a formula is of no practical value, and is riddled with special cases. In spite of these complications, Mathematica can solve such systems of equations with numerical coefficients, both in exact form and approximated numerically. These solutions are very useful in the study of conic curves introduced in later chapters.

Example: Find approximate numerical solutions for the system of equations

using the Mathematica NSolve command.

Solution: The Mathematica function NSolve[eqnList,varList] returns a list of numerical solutions for a system of equations in several variables. The solution(s) are returned in the form of Mathematica rules.

Clear[x,y];

NSolve[{3x^2+2x*y-4y^2-2x-3y-4==0,

x^2-4x*y+y^2+3x+4y+1==0},{x,y}]

Notice that in this example two of the solution pairs involve only real numbers, and two involve complex numbers. The complex solutions are a conjugate pair.

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Descarta2D Hint: Descarta2D provides the function Solve2D to supplement the capabilities of the Mathematica Solve function. It provides specialized capabilities that are useful in the implementation of the Descarta2D packages. Refer to the Descarta2D references for a detailed description of the Solve2D function.

Graphs [Top]

Consider that F(x,y)=0 has been solved for y so that y=f(x). We wish to give a geometric interpretation to the equation y=f(x). Now if a value, say , is assigned to x, then, if f(x) is single-valued, there will be determined a single value y, say . Another value of x, say , will produce a value . If f(x) is multiple-valued, there will be several values of y for a given x. In any event the real number pairs which satisfy y=f(x) may be plotted in two dimensions as points in the plane. The aggregate of these points constitutes the graph or plot of the equation y=f(x) or of the function f(x).

This is one of the central problems in plane analytic geometry: given a function y=f(x), to plot its graph or to represent it geometrically. We sometimes say that the graph of f(x) is the locus of f(x). The word locus, in general, carries with it the idea of motion. Thus, the curve traced by a moving point is called the locus of the point. Such a locus is also referred to as a curve in the plane.

Through the study of equations much can be learned about the geometric properties of graphs. Such analysis is one of the roles of analytic geometry. In the study of an equation y=f(x) there are many analyses that can be made in order to intuitively understand the behavior of the graph. Mathematica and Descarta2D can be used to aid in this understanding. Four properties of significant interest in analytic geometry are

Intercepts The points at which the curve crosses the x- and y-axes.

Extent The regions of the plane to which the curve is confined and regions where it tends to infinity.

Symmetry The lines in which the reflection of the curve is a mirror image of the curve itself. Cases of interest include symmetry about the x- or y-axes, symmetry about the origin, and symmetry about the lines y=x or y=-x.

Asymptotes The behavior of an unbounded curve in the neighborhood of infinity, where either x, y, or both become infinite. In particular, it may happen that the distance from a point P on the curve to some fixed line tends to zero. Such a line is called an asymptote of the curve.

The set of all points which satisfy a given condition is called the locus of that condition. An equation is called the equation of the locus if it is satisfied by the coordinates of every point on the locus and by no other points. There are three common representations of the locus by means of equations:

Rectangular equations which involve the rectangular coordinates (x,y)

Polar equations which involve the polar coordinates (r,θ)

Parametric equations which express x and y (or r and θ) in terms of a third independent variable called a parameter.

This book focuses on rectangular and parametric equations, with polar equations covered in the explorations.

Parametric Equations [Top]

It is often advantageous to use two equations to represent a curve instead of one. The x-coordinate of a point on the curve will be given by one equation expressing x as some function of a parameter, say θ or t, and the y-coordinate will be given by another equation expressing y as a function of the same parameter. Such equations are called parametric equations. Upon eliminating the parameter between the two equations the implicit equation, in the form f(x,y)=0, of the curve may be found. Some loci problems are treated most readily by means of parametric equations. Parametric equations are also the most natural means for generating a sequence of points on a curve, such as those needed to plot the curve. Since a parameter may be chosen in many ways, the parametric equations of a given curve are not unique, and in some cases they will only represent a portion of a curve.

Example: Find parametric equations of the locus of a point as it "orbits" about the origin at a distance of 2 units.

Solution: Let the parameter θ be the angle measured counter-clockwise from the +x-axis that a line segment of length 2 sweeps when anchored at the origin (0,0). Using trigonometry the x- and y-coordinates of the end point of the line segment are given by the parametric equations

x=2sinθ and y=2cosθ.

The locus of these parametric equations is a circle. In Mathematica a parametric curve may be plotted using ParametricPlot[{x(t),y(t)},{t,t,t}] where x(t) and y(t) are the parametric equations of the curve, t is the parameter, and t and t are the start and end values of the parameter.

Clear[t];

ParametricPlot[{2Sin[t],2Cos[t]},{t,0,2Pi},

AspectRatio->Automatic]

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In our study of curves in the plane we will examine both implicit and parametric equations for the curves.

Explorations [Top]

Determinants. [deter.html]

Determinants often provide a concise notation for expressing relationships in analytic geometry. Show that the expanded algebraic form for the 2×2 determinant

is given by . Show that the expanded algebraic form for the 3×3 determinant

is given by .

Cramer's Rule (Two Equations). [cramer2.html]

Show that the solution to the system of two linear equations in two unknowns

= | 0 | |

= | 0 |

is given by the determinants

where

Cramer's Rule (Three Equations). [cramer3.html]

Show that the solution to the system of three linear equations in three unknowns

= | 0 | |

= | 0 | |

= | 0 |

is given by the determinants

where

Polar Equations. [polareqn.html]

A curve in polar coordinates may have more than one equation. A given point may have either of two general coordinate representations

. | (r,θ+2kπ), | . |

. | (-r,θ+(2k+1)π), | . |

for any integer k. Hence a given curve r=f(θ) may have either of the two equation forms

r | = | f(θ+2kπ), |

-r | = | f(θ+(2k+1)π). |

The first equation reduces to r=f(θ) when k=0, but may lead to an entirely different equation of the same curve for another value of k. Similarly, the second equation may yield other equations of the curve. Show that in spite of the potential for multiple equations in polar coordinates, a linear equation Ax+By+C=0 has only one representation in polar coordinates given by

r(Acosθ+Bsinθ)+C=0.

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