Text two analytic geometry

The rectangular coordinate system provides one-to-onecorrespondence between number pairs and points; that is,  corresponding to a number pair( ) there is always one and only one point ; and corresponding to a point there is one and only one number pair( ).  This one-to-one correspondence is the starting point of the plane Analytic Geometry.

The notion of a correspondence between a point in the plane and a pair of numbers can extended to a more general kind of correspondence,  namely,  between a geometric locus and an equation.  The graph of an equation is the locus of the points whose coordinates satisfy the equation. Conversely the equation of given curve is an equation satisfied by the coordinates of every point on the curve and by thecoordinates of no other points.

This correspondence between equations and geometric loci, will indeed, form the central subject of our study. That is to say, our main investigation will take the form of one or the other of the problems:

1.  Given an equation, to obtain the corresponding geometric locus(the graph of the equation)  along with its properties

2.  Given a geometric locus whose points some common property(shared by no other points), to  find the corresponding equation.

In the latter case the equation, in turn, will help us in studying other properties of the locus.

Thus,we define a curve as composed of points whose coordinates satisfy a certain equation.  We may think of a curve as a locus or a path traced by a moving point according to certain specified conditions. From these conditions it is possible to derive the equation of its curve and then discuss the curve in detail from the equation. The locus of an equation in X and Y is defined as the totality of points whose coordinates satisfy the equation.  There exists no definite rule for finding the equation of the locus.  As a matter of fact the problem is to translate the geometric definition of the locus into an algebraic form with a suitable choice of a coordinate system.

We shall proceed to the discussion of particular species of loci-namely, the straight line, a circle,  a parabola,  an ellipse,  and a hyperbola.

The problem of finding the equation of the straight line is the simplest case of the general problem of finding the equation of a curve.  The equation of a straight line is determined by two points P( , ) and ( , ). This equation will be obtained from the fact that the point P(X,  Y)  is on the straight line,  if and only if,  the slopes of the segments Pand are equal. This condition is (Y- )/( - )=( - )/( - ),  ≠ .We shall refer to this as the two point form of the equation of the straight line. Thus any straight line may be represented by an equation of the first degree in X andY. Conversely, every equation of first degree Ax+ By+C=0 represents a straight line.

The following loci lead to particular type of second degree equations, in two variables. 

The Circle is the locus of a point, which moves so that its distance from a fixed point, called a centre,  is constant.  The distances from itscentreto the locus are radii of the circle.  Thus,  is the equation of the circle with the centre at the origin and a radius r.

The Parabola is the locus of points which are equidistant from a fixed point and a fixed straight line.

  The fixed point is the focus, the fixed line is the direotrix.  The line perpendicular to the directrix and passing through the focus is the axis of the parabola.  The axis of the parabola is, obviously, a line of symmetru.  The point on the halfway between the focus and the directrix on the parabola is the vertex of the parabola.The parabola is fixed when the focus and the directrix are fixed.  The equation of the parabola, however, depends on the choice of the coordinate system.If the vertex of the parabola, is at the origin and the focus is at the point (O,P)  its equation is =2PY or =2PX.

The Ellipse-is the locus of a point which moves so that the sum of its distances from two fixed points called the foci is constant.  This constant will be denoted by 2a, which is necessarily greater than the distance between the foci(the focal distance).  The line through the foci is the principal axis of theellipse, the points in which the ellipse cuts the principal axis are called the vertices of the ellipse. If the centre of the ellipse is at the origin but the foci are on the y-axis its equation is

,

where a and b represent the lengths of its semimajor and semiminor axes.

The Hyperbola is the locus of a point which moves so that the diffierence of its distances from two fixed points is a constant lion is 2a.  Its equation is

.

This equation shows that the hyperbola is symmetric with respect to both coordinate axes also the origin.  It intersects the X-axis but does not cut the Y axis.  Hence, the curve is not contained in a bounded portion of a plane.  The curve consists of two branches.  The line segment joining the vertices is called the transverse axisof the hyperbola; its length 2a.  The point midway between the vertices is a geometrical centre and is called the centre of the hyperbola.

  Read and translate the text in class.  Discuss the problems involved.