Term Paper Coordinate Geometry


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Term Paper Coordinate Geometry

  2. 2. Acknowledgement History of all great works is to witness that no great work was ever done without either the active or passive support a person‘s surrounding and one’s close quarters. Thus it is not hard to conclude how active assistance from seniors. Could prohibitively impact the execution of a project .I am highly thankful to our learned faculty Mr. Pankaj Kumar for her active guidance throughout the completion of project. Last but not the least, I would also want extend my appreciation to those who could not be mentioned here but here well played their role to inspire the curtain. Durgesh Kumar Singh
  3. 3. Contents 1. History 2. Introduction 3. Coordinates of mid point 4. Distance between two points 5. Section formula 6. Coordinates of centroid 7. Area of triangle 8. Straight lines 9. Slope of straight lines 10. General equation of straight line 11. Angle between two straight lines
  4. 4. 12. Length of perpendiculars from a point 13. Uses of coordinate geometry
  5. 5. History The Greek mathematician Menaechmus, solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had analytic geometry.[1] Apollonius of Perga, in On Determinate Section dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others.[2] Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different than our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves. However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve a posteriori instead of a priori. That is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation.[3] The eleventh century Persian mathematician Omar Khayyám saw a strong relationship between geometry and algebra, and was moving in the right direction when he helped to close the gap between numerical and geometric algebra[4] with his geometric solution of the general cubic equations,[5] but the decisive step came later with Descartes.[4] Analytic geometry has traditionally been attributed to René Descartes[4][6][7] who made significant progress with the methods of analytic geometry when in 1637 in the appendix entitled Geometry of the titled Discourse on the Method of Rightly Conducting the Reason in the Search for Truth in the Sciences, commonly referred to as Discourse on Method. This work, written in his native French tongue, and its philosophical principles, provided the foundation for calculus in Europe. Abraham de Moivre also pioneered the development of analytic geometry. With the assumption of the Cantor-Dedekind axiom, essentially that Euclidean geometry is interpretable in the language of analytic geometry (that is, every theorem of one is a theorem of the other), Alfred Tarski's proof of the decidability of the ordered real field could be seen as a proof that Euclidean geometry is consistent and decidable
  6. 6. Introduction to Coordinate Geometry from Latin: coordinate "to set in order, arrange" Coordinate geometry is that branch of geometry in which two real numbers called coordinates are used to indicate the position of point in a plane. The main contribution of coordinate geometry is that it has the fact that it has inabled the integration of alzebra and geometry. This is an evident from the fact that alzebric methods are employed to represent and prove the fundamental properties of geometrical theorems. Equations are also employed to represent the various geometric figures. It is because of these features that the coordinate geometry is considerd to be most powerful tool to analysis than the Euclidian geometry. It is on this consideration that sometimes it describes as analytical geometry. Directed lines:- a directed line is a straight line with number units positive , zero and negative. The point of origin is number 0. The arrow indicates its direction . on the side of arrow are the positive numbers and on the other side are the negative numbers. It is like a real number scale illustrated below: A directed line can be horizontal normally indicated by XOX’ axis and vertical indicated normally by YOY’ axis. The point where the two intersects is called point of origin. The two lines together are called rectangular axis and are at right angle to each other. If these axes are noy at right angles they are said to be oblique axis and the angle between the positive axes XOY is denoted by ω (omega).
  7. 7. What are coordinates? To introduce the idea, consider the grid on the right. The columns of the grid are lettered A,B,C etc. The rows are numbered 1,2,3 etc from the top. We can see that the X is in box D3; that is, column D, row 3. D and 3 are called the coordinates of the box. It has two parts: the row and the column. There are many boxes in each row and many boxes in each column. But by having both we can find one single box, where the row and column intersect. The Coordinate Plane In coordinate geometry, points are placed on the "coordinate plane" as shown below. It has two scales - one running across the plane called the "x axis" and another a right angles to it called the y-axis. (These can be thought of as similar to the column and row in the paragraph above.) The point where the axes cross is called the origin and is where both x and y are zero.
  8. 8. On the x-axis, values to the right are positive and those to the left are negative. On the y-axis, values above the origin are positive and those below are negative. A point's location on the plane is given by two numbers, one that tells where it is on the x-axis and another which tells where it is on the y-axis. Together, they define a single, unique position on the plane. So in the diagram above, the point A has an x value of 20 and a y value of 15. These are the coordinates of the point A, sometimes referred to as its "rectangular coordinates". For a more in-depth explanation of the coordinate plane see The Coordinate Plane. For more on the coordinates of a point see Coordinates of a Point The coordinate plane is a two-dimensional surface on which we can plot points, lines and curves. It has two scales, called the x-axis and y-axis, at right angles to each other. The plural of axis is 'axes' (pronounced "AXE-ease"). X axis The horizontal scale is called the x-axis and is usually drawn with the zero point in the middle. As you go to the right on the scale the values get larger and are positive. As you go to the left, they get larger but are negative. Y axis The vertical scale is called the y-axis and is also usually drawn with the zero point in the middle. As you go up from zero the numbers are increasing in a positive direction. As you go down from zero they increase but are negative. Origin The point where the two axes cross (at zero on both scales) is called the origin. In the figure above you can drag the origin point to reposition it to a more suitable location at any time. Quadrants When the origin is in the center of the plane, they divide it into four areas called quadrants. The first quadrant, by convention, is the top right, and then they go around counter- clockwise. In the diagram above they are labelled Quadrant 1,2 etc. It is conventional to label them with numerals but we talk about them as "first, second, third, and fourth quadrant".
  9. 9. In the diagram above, you can drag the origin all the way into any corner and display just one quadrant at a time if you wish. Things you can do in Coordinate Geometry If you know the coordinates of a group of points you can: • Determine the distance between them • Find the midpoint, slope and equation of a line segment • Determine if lines are parallel or perpendicular • Find the area and perimeter of a polygon defined by the points • Transform a shape by moving, rotating and reflecting it. • Define the equations of curves, circles and ellipses. Information on all these and more can be found in the pages listed below. Coordinates of midpoint:- we can find out the coordinates of a mid-point from the coordinates of the any two points using the following formula : Xm = X1+X2 , Ym = Y1+Y2 2 2 For example, the coordinates of the midpoint of the join of points (-2,5),(6,3) are ( -2+6/2 , 5+3/2 ) , ie ., (2,4) This is helpful first in finding out the middle point from the joint of any two points and secondly in verifying whether two straight lines bisect each other.
  10. 10. In the diagram , the dotted vertical lines are drawn perpendicular to x-axis and the dotted horizontal line are parallel to the x-axis. The ΔNMP and ΔQml are the congruent triangles. It follows , therefore , that NM=ML BC=CD OC-OB=OD-OC (Xm-X1) = (X2-Xm) Xm = (X1+X2)/2 …………………….(1) Also from the same congruent triangles, we get NP=QL NB-PB=QD-LD MC-PB=QD-MC Ym-Y1=Y2-Ym Ym=(Y1+ Y2)/2 ……………………………….(2) From equation 1 and 2 , we coclude that the coordinates of the midpoint (Xm,Ym) are ( (X1+X2)/2 , (Y1+Y2)/2 ) .
  11. 11. Distance between two points :- The distance , say d, between two points P(x1,y1) and Q(x2,y2) is given by the formula d= √ (x2-x1)² + (y2-y1)² since we take square of the two diffrences , we may get designate any mof the points as (x1,x2) and the other (x2,y2). In order to prove above formula , let us take two points P(x1,y1) and Q(x2,y2) as shown in following diagram.
  12. 12. The vertical dotted lines PB and QC are perpendicular from P and Q on the X axis and PR is the perpendicular from P and Qc. Then PR=BC=OC-OB=x2-x1 And QR=QC-RC=y2-y1 From the right angled triangle PRQ, right angled at R, we have by Pythagoras theorem PQ2=PR2+QR2 d2 = (x2-x1)2+(y2-y1)2 d = √(x2-x1)2+(y2-y1)2
  13. 13. SECTION FORMULA:- The coordinates of point R(x,y) dividing the line in the ratio of m:n connecting the points P(x1,y1) and Q(x2 , y2) in the diagram below x= and y= Draw PL, RM and QN perpendiculars on XOX’ and draw PK and RT perpendiculars on RM and QN respectively. We are given
  14. 14. = Suppose PQ makes angle Ѳ with the X-axis .from the figure: In ∆PRK, CosѲ PR cosѲ =x-x1 and in ∆RQT RQ cos Ѳ =x2-x Dividing (1) by (2) we get nx-nx1=mx2-mx x= similarly
  15. 15. ny-ny1=my2-my y (m+n)=my2+ny1 y= it may be noted that m which corresponds to the segment PR multiplies the coordinate of Q, while n, which refers to RQ, multiplies the coordinate of P. if m=n we find that the coordinate of the midpoint are
  16. 16. External division In the above it is assumed that the point R divides PQ internally in the ratio m: n. This means that the point R lies between P and Q. if PQ is divided externally by R, then R lies outside PQ. The student should repeat the above method and using the same diagram with R and Q interchanged, it can be proved that X= y= Coordinates of centroid The centroid of a triangle is the point of intersection of the three medians of triangle. Each median bisects the side opposite to the vertex into two equal parts. In order to prove that the median of triangle intersects at a point called centroid, we have to showthat the coordinates are X= and y= To prove this let us take a triangle with its vertices A(x1,y1), B(x2,y2) and C(x3,y3) as shown in the following diagram:
  17. 17. In the diagram median AD bisects the base BC at D with coordinates D= ) we take a point G where no medians intersect which divides AD internally , say, in ratio 2:1 ,i.e.,m1:m2 = 2:1 hence by the section formula the coordinates of G are X= And Y= Area of triangle
  18. 18. We can find out the area of the triangle with the vertices given. For this let A(x1,y1), B(x2,y2) and C(x3,y3) be the coordinates of the vertices of the triangle ABC. From A,B,C draw the perpendiculars AL, BM and CN on the x-axis. As is clear from the figure area of ∆ABC = area of trapezium ABML+area of trapezium ALNC – area of trapezium BMNC Since the area of the trapezium = [sum of the parallel sides] × [perpendicular distance between the parallel sides] Hence the area of the ∆ABC can be given as ∆ABC = [BM+AL]ML + (AL+CN)LN - (BM+CN)MN ∆ABC = ] Note: - if the three points are collinear then the area of ∆ABC is equal to zero The straight line
  19. 19. The study of curve starts with the straight line which is simplest geometrical entity. Mathematically it is defined as the shortest distance between the two points. Slope of a straight line Slope of the line is the tangent of the angle formed by the line above the x-axis towards its positive direction, whatever be the position of the line as shown below : Slope of a line is generally denoted by m . thus if a line makes an angle Ѳ with the positive direction of the x-axis , its slope is m=tanѲ if Ѳ is acute , slope is positive and if Ѳ is obtuse , the slope is negative . in terms of the coordinates , the slope of the line joining two points , say A(x1,y1) and B(x2,y2) is given by: m= tanѲ= The following diagrams will make the explanation more clear
  20. 20. tanѲ = DIFFERENT FORMS OF EQUATIONS OF THE STRAIGHT LINE 1.Equations of coordinate axes: If P(x,y) be the point on the x-axis , then its ordinate y is always zero for any position of the point P on the x-axis and for no other point. ... y=o Is the equation of x axis. If P(x,y) is any point on the y-axis , then its abscissa x is always zero for any position of the point P on the y-axis and for no other point. ... x=0, is the equation of y-axis.
  21. 21. Equation of lines parallel to the coordinate axis:- Let P(x,y) be any point on a line parallel to y-axis at a distance ‘a’ from it . for any position of the point P lying on this line and for no other point the abscissa x is always constant and is equal to a. ... x=a is the equation of the line parallel to the y-axis and at a distance a from it. Similarly y=b is the equation of the line parallel to the x-axis and at a distance b from it .
  22. 22. Origin slope form The eq. of line passing through the origin and having slope m: Let a straight line pass through the origin o and have a slope m. let P(x,y) br any point on the line. From P draw PM perpendicular on the x-axis then, y=xtanѲ y=mx which is the required eq. of the line.
  23. 23. A line intercepting the axes :- In case the straight line meets the x-axis and the y-axis at points other than the origin, the respective points will be called x-intercept and y-intercept. The diagram shows the two intercepts of a straight line AB which meet the x-axis in A and the y-axis in B, the OA is called the intercept of the line on the y-axis , and the two intercept OA and OB taken in the particular order are called the intercepts of the line on the axes. It may be noticed that at the point of y intercept , the value of x is equal to zero and at the point of x-intercept, the value of y is equal to zero. Therefore in order to find out the value of say y-intercept we have to put x equal to zero in the equation. Similarly to find out the value of x-intercept , we put y equal to zero in the equation.
  24. 24. Slope intercept form:- The eq. of line with the slope m and an intercept c on y-axis. Let a straight line of slope m intersects the y axis in K. let OK the intercept on the y-axis , be c . then the coordinates of K are (0,c). Take P(x,y) any variable point on the line. ... Slope of KP = …………(1) Slope of line as given in m …………….(2) Equating eq. (1) and (2) , we get the required equation as y=mx+c
  25. 25. Parametric Form In the figure if the point P(x,y) is taken at a distance r from the point r from the point R on the line , then ... is the required eq. of the straight line in parametric form.
  26. 26. Two point form:- The eq. of straight line passing through two points (x1,y1) and (x2,y2) Let Q(x1,y1) and R(x2,y2) be the two points on the line. Take any point P(x,y) on the line. Then by def., Slope of QP= ………..(1) Slope of QR= …………(2) Since A,P,B are collinear points from (1) and (2) , we have Hence the required fro straight line is y-y1 = Normal perpendicular form:-
  27. 27. The eq of straight line in terms of the perpendicular form the origin and the inclination of the perpendicular with the axis: Let a straight line be at a perpendicular distance p from the origin and the inclination of the perpendicular OM with the x-axis be α. The coordinates of M , the foot of the perpendicular , in terms of the given constants are (p cos α , p sin α ). The inclination of the line with the x-axis is Slope of the given line = tan let P(x,y) be any point on the line , then slope of MP= since A,P,B are collinear from(1) and (2), we have is the required equation.
  28. 28. General equation of a straight line An equation of the form ax+by+c=0, where a, b, c are constants and x, y are variables, is called the general equation of the straight line. Slope of the line ax+by+c=o m=-
  29. 29. The angle between the two straight lines Let AB and CD be two straight lines with given inclination to the x-axis as Ѳ1 and Ѳ2, form interior and exterior angles Ѳ and α respectively as shown below: Since Ѳ1=Ѳ+Ѳ2 Ѳ=Ѳ1-Ѳ2 ……………(1) Also ∏=Ѳ+α ... α=∏-Ѳ ………(2) 1. For the interior angle: tanѲ=tan(Ѳ1-Ѳ2) if we express the slopes in terms of m1 and m2 , then the formula can also be expressed as
  30. 30. tanѲ= 2.For exterior angle, therefore tanα=tan(∏-Ѳ) = -tan Ѳ = - Condition Of Parallelism If two line are parallel, the angle between them is zero degree. m1-m2=0, i.e., m1=m2 Condition Of Perpendicularism If the two lines are perpendicular Then tanѲ=tan90˚=∞
  31. 31. so m1m2=-1 Length of perpendicular from a point
  32. 32. Let AB be the line ax+by+c=0 ……………(1) let P(x1,y1) be the points and PM perpendicular to AB. Join AP and PB . AB meets the x-axis , where putting y=0 in (1), we get ax+c=0 or x=- ... coordinate of A is Similarly coordinate of B is (0,-
  33. 33. So the length of perpendicular= Uses of coordinate geometry in daily life In construction of buildings In determining length between two points In finding slope of any thing. Ordanance survey Surveying
  34. 34. References Mathematics (N.C.E.R.T) Business mathematics (S.D.Sharma) www.wikipedia.org www.mathstutor.com www.mathopenref.com