Pre – CalculusBSMath 5-2iiAcknowledgementContents1 Analytic Geometry 1 1.1 Introduction to Conic Sections and Circles . .

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.21.2 Parabola . . .

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. .41.3 Hyperbola .

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. .91.4 Systems of Nonlinear Equations . .

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. .14iv CONTENTSUnit 1Analytic GeometryIntroductionThe study of the geometry of gures by algebraic representation andmanipulation of equations describing their positions, congurations, andseparations is called Analytic Geometry .Analytic geometry is alsocalled coordinate geometry since the ob jects are described as n-tuplesof points (where n=2 in the plane and 3 in space) in some coordinatesystem.2 Analytic Geometry1.1 Introduction to Conic Sections andCirclesConics and CirclesConic sections are the curves which can be derived from taking slices of a”double-napped” cone. (A double-napped cone, in regular English, is twocones “nose to nose”, with the one cone balanced perfectly on the other.

)”Section” here is used in a sense similar to that in medicine or science,where a sample (from a biopsy, for instance) is frozen or suused witha hardening resin, and then extremely thin slices (“sections”) are shavedo for viewing under a microscope. If you think of the double-nappedcones as being hollow, the curves we refer to as conic sections are whatresults when you section the cones at various angles. A circle is a geometrical shape, and is not of much use in algebra, sincethe equation of a circle isn’t a function. But you may need to work withcircle equations in your algebra classes. In “primative” terms, a circle isthe shape formed in the sand by driving a stick (the “center”) into thesand, putting a loop of string around the center, pulling that loop tautwith another stick, and dragging that second stick through the sand atthe further extent of the loop of string. The resulting gure drawn in thesand is a circle. In algebraic terms, a circle is the set (or “locus”) of points (x, y) atsome xed distance r from some xed point (h, k).

The value of r is calledthe “radius” of the circle, and the point (h, k) is called the “center” ofthe circle. The “general” equation of a circle is:x2+ y2+ Dx +E y +F = 0The “center-radius” form of the equation is: (x h)2+ ( y k)2= r2where the h and the k come from the center point (h, k) and the r2 comesfrom the radius value r. If the center is at the origin, so ( h; k) = (0 ;0),then the equation simplies to x2+ y2= r2.

You can convert the “center-radius” form of the circle equation intothe “general” form by multiplying things out and simplifying; you canconvert in the other direction by completing the square. The center-radius form of the circle equation comes directly from theDistance Formula and the denition of a circle. If the center of a circleis the point ( h; k) and the radius is length r, then every point (x, y) onthe circle is distance r from the point ( h; k). Plugging this information1.

1 Introduction to Conic Sections andCircles 3 into the Distance Formula (using r for the distance between the pointsand the center), we get:r= p (x h)2+ ( y k)2r 2= p (x h)2+ ( y k)2 r 2= ( x h)2+ ( y k)2Properties of Circle Circles having equal radii are congruent. Circles having dierent radii are similar. The central angle which intercepts an arc is the double of any inscribedangle that intercepts the same arc.

The radius perpendicular to a chord bisects the chord. The chords equidistant from the center are equal in length. A tangent to a circle is at a right angle to the radius at the point ofcontact.

Two tangents drawn on a circle from a point outside are equal in length. The angle subtended at the center of a circle by its circumference isequal to four right angles. The circumference of two dierent circles is proportional to their cor-responding radii.

Arcs of the same circle are proportional to their corresponding angles. Radii of the same circle or equal circles are equal. Equal chords have equal circumferences.

The diameter of a circle is the longest chord.4 Analytic Geometry1.2 ParabolaA parabola (plural “parabolas”; Gray 1997, p. 45)is the set of all points in the plane equidistant froma given line (the conic section directrix) and a givenpoint not on the line (the focus). The focal param-eter (i.e.

, the distance between the directrix and fo-cus) is therefore given by , where is the distance fromthe vertex to the directrix or focus. The surface ofrevolution obtained by rotating a parabola about itsaxis of symmetry is called a paraboloid. The parabola was studied by Menaechmus in anattempt to achieve cube duplication. Menaechmussolved the problem by nding the intersection ofthe two parabolas and . Euclid wrote about theparabola, and it was given its present name by Apol-lonius. Pascal considered the parabola as a pro-jection of a circle, and Galileo showed that pro jec-tiles falling under uniform gravity follow parabolicpaths.

Gregory and Newton considered the cata-caustic properties of a parabola that bring parallelrays of light to a focus (MacTutor Archive), as illus-trated above. For a parabola opening to the rightwith vertex at (0, 0), the equation in Cartesian co-ordinates is:p (x a)2+ y2= x+ a( x a)2+ y2= ( x+ a)2x 2 2ax +a2= x2+ 2 ax+a2y 2= 4 axThe quantity 4 ais known as the lactus rectum. Ifthe vertex is at ( x0; y0) instead of (0;0), the equationof the parabola is:(y y0)2= 4 a(x x0).1.2 Parabola 5Three points uniquely determine one parabolawith directrix parallel to the x-axis and one withdirectrix parallel to the y-axis. If these parabolaspass through the three points ( x1; y1), (x2; y2), and( x3; y3) they are given by equations: x2x y 1x 21 x1 y1 1x 22 x2 y2 1x 23 x3 y3 1 = 0and y2x y 1y 21 x1 y1 1y 22 x2 y2 1y 23 x3 y3 1 = 0 In polar coordinates, the equation of a parabola with parameteraand acenter (0 ;0) is given byr= 2a 1 +cos(left gure).

The equivalence with the Cartesian form can be seen bysetting up a coordinate system ( x0; y 0) = ( x a; y ) and plugging inr = p x02+ y02and = tan 1( y0 x0) to obtainp (x a)2+ y2= 2a 1 +x a p(x a)2+ y26 Analytic GeometryExpanding and collecting terms,a + x+ p (a x)2+ y2= 0so solving for y2gives ( }). A set of confocal parabolas is shown in thegure on the rightIn pedal coordinates with thepedal point at thefocus, the equationisp2= ar:The parabola can be written parametrically asx= at2y = 2 atorx= t2 4ay = t:A segment of a parabola is a Lissaious curve A parabola may be generated as the envelope of two concurrent linesegments by connecting opposite points on the two lines (Wells 1991).1.2 Parabola 7In the above gure, the linesS P A; S QB;andP OQ are tangent to theparabola at points A; B;andO;respectively. Then S P P A=QO OP=BQ QS(Wells 1991). Moreover,the circumcircle of P QS passes through the focusF(Honsberger1995, p.

47). In addition, the foot of the perpendicular to a tangentto a parabola from the focusalways lies on the tangent at the vertex(Honsberger 1995, p.48). Given an arbitrary pointPlocated “outside” a parabola, the tangentor tangents to the parabola through Pcan be constructed by drawingthecircle havingP Fas adiameter , whereFis thefocus .Then locatethe points Aand Bat which the circle cuts thevertical tangent throughV .The points TA andTB (which can collapse to a single point in thedegenerate case) are then the points of tangency of the lines P Aand P Band the parabola (Wells 1991).8 Analytic GeometryThecurvature ,arc length , andtangential angle areK (t) = 1 2a (1 + t2) 3 2s (t) = a(tp 1 +t2+ sinh 1t) (t) = tan1t:Thetangent vector of the parabola isx(t) = 1 p1 +t2y (t) = t p1 +t2The plots below show the normal and tangent vectors to a parabola.

1.3 Hyperbola 91.3 HyperbolaA hyperbola (plural “hyperbolas”; Gray 1997,p.

45) is aconic sectiondened as thelocusof allpoints Pin theplanethe dierence of whose dis-tances r1 =F1Pand r2 =F2Pfrom two xed points(thefoci F1andF2) separated by a distance 2cis agivenpositiveconstant k,r 2 r1 =k(Hillbert and Cohn-Vossen 1999, p. 3). Letting Pfall on the left x-intercept requires thatk = ( c+ a) (c a) = 2 a;so the constant is given by k= 2 a,i.e.

, the distancebetween the x-intercepts (left gure above). The hy-perbola has the important property that a ray orig-inating at afocus F1 reects in such a way that the outgoing path liesalong the line from the otherfocusthrough the point of intersection (rightgure above).The special case of therectangular hyperbolacorresponding to a hy-perbola with eccentricity e= p 2 , was rst studied by Menaechmus.

Eu-clid and Aristaeus wrote about the general hyperbola, but only studiedone branch of it. The hyperbola was given its present name by Apol-lonius, who was the rst to study both branches. Thefocusandconicsection directrixwere considered by Pappus (MacTutor Archive). Thehyperbola is the shape of an orbit of a body on an escape tra jectory (i.e.,a body with positive energy), such as some comets, about a xed mass,such as the sun. The hyperbola can be constructed by connecting the free endXof arigid bar F1X, where F1 is afocusand the otherfocusF2 with a stringF 2P X.

As the bar AXis rotated about F1 andPis kept taut againstthe bar (i.e., lies on the bar), thelocusof Pis one branch of a hyperbola(left gure above; Wells 1991).

A theorem of Apollonius states that for a10 Analytic Geometryline segment tangent to the hyperbola at a pointTandintersectingtheasymptotes at points Pand Q, then OP OQis constant, and P T=QT(right gure above; Wells 1991). Let the pointPon the hyperbola have Cartesian coordinates( x; y), thenthe denition of the hyperbola r2 r1 = 2agivesp (x c)2+ y2 p (x + c)2+ y2= 2 aRearranging and completing the square givesx2(c 2 a2) a2y 2= a2(c 2 a2);and dividing both sides by a2(c 2 a2) results inx 2 a2 y2 c2 a2= 1:By analogy with the denition of theellipse,deneb2= c2 a2;so the equation for a hyperbola withsemima jor axis aparallel to thex-axisandsemiminor axis bparallel to they-axisis given byx2 a2 y2 b2= 1:or, for a center at the point x0; y0) instead of (0;0),( x x0)2 a2(y y0)2 b2= 1:1.3 Hyperbola 11Unlike theellipse, no points of the hyperbola actually lie on thesemimi-nor axisbut rather the ratio b adetermines the vertical scaling of thehyperbola. Theeccentricity eof the hyperbola (which always satisese ; 1 ) is then dened ase= c a= q 1 +b2 a2:In the standard equation of the hyperbola, the center is located at ( x0; y0), thefociare at x0 c; y0) , and the vertices are atx0 a; y0). The so-calledasymptotes(shown as the dashed lines in the above gures) can befound by substituting 0 for the 1 on the right side of the general equationy= b a(x x0) +y0;and therefore haveslopes b athe special case a= b(the left diagram above) is known as arectangularhyperbolabecause theasymptotesareperpendicular.

The hyperbola can also be dened as thelocusof points whose distancefrom thefocus Fis proportional to the horizontal distance from a ver-tical line Lknown as theconic section directrixwhere the ratio is