1
           Chapter1
           Straight Line
           1.1  Introduction
           The concept of analytical geometry was given by Percey Smith and Arthur
           Gale. Percey Smith was a professor of mathematics in Sheffield scientific
           school, Yale university and Arthur Gale was an assistant professor of math-
           ematics in the university of Rochester. Analytic geometry is widely used in
           physics, engineering, aviation, rocketry, space science, and spaceflight. It is
           the foundation of most modern fields of geometry, including algebraic, dif-
           ferential, discrete and computational geometry. Usually the cartesian coor-
           dinate system is applied to manipulate equations for planes, straight lines,
           and squares, often in two and sometimes in three dimensions. Geometri-
           cally, one studies the Euclidean plane (two dimensions) and Euclidean space
           (three dimensions). As taught in school books, analytic geometry can be ex-
           plained more simply: it is concerned with defining and representing geomet-
           rical shapes in a numerical way and extracting numerical information from
           shapes’ numerical definitions and representations. That the algebra of the
           real numbers can be employed to yield results about the linear continuum of
           geometryrelies on the Cantor–Dedekind axiom.
             Analytic geometry is widely used in physics and engineering, and also
           in aviation, rocketry, space science, and spaceflight. It is the foundation of
           mostmodernfieldsofgeometry,includingalgebraic,differential,discreteand
           computational geometry. Usually the cartesian coordinate system is applied
           to manipulate equations for planes, straight lines, and squares, often in two
           andsometimesinthreedimensions. Geometrically,onestudiestheEuclidean
           plane(twodimensions)andEuclideanspace(threedimensions). Astaughtin
           schoolbooks,analyticgeometrycanbeexplainedmoresimply: itisconcerned
           with defining and representing geometrical shapes in a numerical way and
       2                      Chapter1. Straight Line
       extracting numericalinformationfromshapes’numericaldefinitionsandrep-
       resentations. That the algebra of the real numbers can be employed to yield
       resultsaboutthelinearcontinuumofgeometryreliesontheCantor–Dedekind
       axiom.
        Analyticgeometryandalgebrabotharesimilar,theyareusedtodesignthe
       geometric objects in the mathematics, and these geometric objects are points,
       straight line, and circle etc. In the plane of geometry, the points are repre-
       sented as ordered pair and in case of straight line it is represented as set of
       points which satisfy the linear equation and the analytic geometry which sat-
       isfies linear equation are known as linear algebra. There are some different
       names for analytic geometry like cartesian geometry is also known as ana-
       lytic geometry. This plane analytic geometry is based on the coordinate sys-
       tem and the principal of algebra and analysis. Now, we will see how to find
       the distance and angle in analytic geometry. Suppose we have coordinates
       m1, m2, n1, n2 for the plane of geometry, then we will see how to find the
       distance of plane of analytic geometry by using above coordinates:
                 D=q(m1−m2)2+(n1−n2)2
       Where D is distance of analytic geometry and the angle of geometry is equal
       φ = tan−1 p, where p is the slope of line. Using this formula, we can find the
       distance of plane of analytic geometric.
        Before understandinganalytical geometry first becomes familiar with his-
       tory of analytical geometry. The analytical geometry history starts from the
       period when a great mathematician of Greece whose name was Menaechmi
       started solving someoftheproblems. Hestartsprovingsomeofthetheorems
       byutilizing a method which was very similar to the utilization of the coordi-
       nates. Thehistoryofanalyticalgeometry,alsoincludesthecontributionofthe
       ApolloniusbelongingtothePergawhoalsosolvedsomeoftheproblemsina
       waythatcanbereferredastheanalyticalgeometryofkindoftheonedimen-
       sionandthequestionwhichtheyusedtosolvenormallywasdeterminingthe
       points which exist on a line and that happened to be in the form of a ratio
       withtheothers. Thefurtherdevelopmentoftheanalyticalgeometrywasalso
       credited to the Apollonius who in the conics again gave a process to solve the
       problemswhichwasverymuchliketheanalyticalgeometry. Hegavehiscon-
       tribution in the fields of the lines of the reference also gave the uses of tangent
       anddiameterwhichweverynormallyuseinthemodernmathinthesystem
       of coordinate in which we find the distance along any diameter starting from
       the point which is called the point of tangency.
              1.2. Point-Slope Form                                                                3
              1.2     Point-Slope Form
              Definition1.1. Astraightlineisrepresentedbyanequationofthefirstdegreeintwo
              va-riables. Conversely, the locus of an equation of the first degree in two variables is
              a straight line. A straight line is completely determined if its direction is known and
              a point is given through which the line must pass.
              Definition1.2. Theequationofthestraightlinethroughpoint P(x1,y1)whoseslope
              is m is
                                              y−y1 =m(x−x1)
              Definition 1.3. The equation of the straight line having slope m and y−intercept
              (0,b) is
                                                   y = mx+b
              Definition 1.4. The equation of the straight line through points P (x ,y ) and
                                                                                      1  1   1
              P (x ,y ) is
                2  2  2                         y−y        y −y
                                                     1 = 1       2
                                                x−x1       x1 −x2
              Definition1.5. Theequationofthestraightlinewhose x and y interceptsarerespec-
              tively (a,0) and (0,b) is             x    y
                                                    a + b = 1
              1.3     GeneralForm
              Definition 1.6. Every equation of the first degree in x and y may be reduced to the
              form
                                               Ax+By+C=0,
              where A, B and C are arbitrary constants, Where the slope m = −A and y-intercept
                   −C                                                            B
              b = B .
              Example1.3.1. Write the equation of the line :
              (a) passing through (−4,3) with slope 1/2.
              (b) passing through (0,−5) with slope −2.
              (c) passing through (2,0) with slope 3/4.
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           Solution:
           Let P(x,y) be any other point on each line. Employ
                                  y−y1 =m(x−x1)
           (a)
                                     y−3=1(x+4)
                                            2
                                    2y−6=x+4
                                x−2y+10=0
           (b)
                                    y−5=−2(x−0)
                                    y−5=−2x
                                2x+y−5=0
           (c)
                                    y−0=3/4(x−2)
                                      4y = 3x−6
                               3z−4y−6=0
           Example1.3.2. Determinetheequationofthelinepassingthroughpoints (−2,−3)
           and (4,2).
              Solution:
           Since                   y−y    y −y
                                       1 = 1   2
                                   x−x1   x1 −x2
           Then
                                     y+3 = −3−2
                                     x+2    −2−4
                                 5x−6y−8=0
           Example 1.3.3. Write the equation of the line whose intercepts on the x and y axes
           are respectively 5 and −3.