Coordinate Geometry two dimensions mathematics

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1. Coordinate Geometry by S. L. Loney

2.   Geometry is concerned with properties of space that are related to distance, shape, size, and the relative position of figures . Algebra is the study of mathematical symbols and the rules for manipulating these symbols appearing in equations describing relationships between variables . René Descartes coupled the branch of geometry with that of algebra, and thereby the study of geometry could be simplified to a very large extent. Of course, this hindered the development of abstract geometry, just like the availability of digital computers blocked the growth of mathematics.

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8. Locus. Equation to a Locus Article 36: When a point moves so as to satisfy a given condition or conditions, the path it traces out is called its locus under these conditions . For example, suppose O is a given point in the plane of the paper and that a point P is to move on the paper so that its distance from O is constant and equal to a. It is clear that all the positions of the moving point must lie on the circumference of a circle whose center is O and radius is a. The circumference of this circle is therefore the "locus" of P when it moves, subject to the condition that its distance from O shall be equal to the constant distance a.

9. Equation to a curve Definition The equation for a curve is the relation that exists between the coordinates of any point on the curve and that holds for no other points except those lying on the curve.

10. Straight Line Article 46      To find the equation for a straight line that is parallel to one of the coordinate axes. Let CL be any line parallel to the axis of y and passing through a point C on the axis of x such that OC = c . Let P be any point on this line whose coordinates are x and y. Then the abscissa of the point P is always c, so that x=c ……..(1)

11. Straight line parallel to the axes Y L P O C

12. Continued The expression (1) is true for every point on the line CL (produced indefinitely both ways), and for no other point is, by article 42, the equation to the line . It should be noted that the equation does not contain the coordinate y . Similarly , the equation for a straight line parallel to the axis of x is y = d . Corollary : The equation to the axis of x is y=0.The equation for the y axis is x = 0.

13. 47. To find the equation to a straight line which cuts off a given intercept on the axis y and is inclined at a given angle to the axis of x. Y L P C N L’ O M X

14. Example

15. 50. To find the equation to the straight line which cuts off given intercepts a and b from the axes. Y B P O M A X

16. 62. To find the equation to the straight line which passes through the two given points (x’, y’) and (x’’, y’’). Y (x’’, y’’) (x’, y’) O X

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21. 66. To find the angle between two given straight lines Y A C2 C1 L2 L1 O X

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24. POINT OF INTERSECTION

25. The circle: Def. A circle is the locus of a point which moves so that its distance from a fixed point, called the centre, is equal to a given distance. The given distance is called the radius of the circle. Y P X M O O M

26. 140. To find the equation to a circle referred to any rectangular axes. Y P O M N X C L

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29. Acknowledgment The author acknowledges all the websites that helped a lot in preparing the slides presented here meant for B.Sc. (Ag) students. D C Agrawal d [email protected] Note : There are a couple of typographical errors; you are supposed to find them out.