COORDINATE GEOMETRY

COORDINATE GEOMETRY

1. Cartesian System

   • René Déscartes, the great French mathematician of the seventeenth century, liked to lie in bed and think! One day, when resting in bed, he solved the problem of describing the position of a point in a plane.
   • His method was a development of the older idea of latitude and longitude. In honour of Déscartes, the system used for describing the position of a point in a plane is also known as the Cartesian system.

2. Plotting a point in the plane if its Coordinates are given

   • Plotting points in a plane (5,4) Step1:- Draw coordinate axis and select the units.  Step 2:- Starting from origin count on units on x and y axis.  Step 3:- Mark the Corresponding points. 

3. CARTESIAN PLANE

   • With plotting graphs on a plane, from the tables of numbers for both linear and non-linear equations. The number line which is also known as a Cartesian plane. 
   • Coordinate Plane Meaning A coordinate plane is a 2D plane which is formed by the intersection of two perpendicular lines known as the x-axis and y-axis.
   • Origin
     • The point at which the axes intersect is known as the origin. 
   • Coordinate axes
     • A coordinate plane is a 2D plane which is formed by the intersection of two perpendicular lines known as the x-axis and y-axis.
     • Cartesian plane is divided into four quadrants by two axes perpendicular to each other, labelled as the x-axis (horizontal line) and the y-axis(vertical line).
     • Abicissa
       • Abscissa is the distance from a point to the vertical or y -axis, measured parallel to the horizontal or x -axis.
     • Ordinate
   • Quadrants
     • The four quadrants along with their respective values are represented in the graph below- Quadrant 1 : (+x, +y) Quadrant 2 : (-x, +y) Quadrant 3 : (-x, -y) Quadrant 4 : (+x, -y)
   • Coordinates of a point
     • Coordinates are a set of values which helps to show the exact position of a point in the coordinate plane.

4. Writing the coordinate of a point

   • We write the coordinates of a point, using the following conventions: (i) The x - coordinate of a point is its perpendicular distance from the y - axis measured along the x -axis (positive along the positive direction of the x - axis and negative along the negative direction of the x - axis). For the point P, it is + 4 and for Q, it is – 6. The x - coordinate is also called the abscissa.
   • (ii) The y - coordinate of a point is its perpendicular distance from the x - axis measured along the y - axis (positive along the positive direction of the y - axis and negative along the negative direction of the y - axis). For the point P, it is + 3 and for Q, it is –2. The y - coordinate is also called the ordinate.
   • (iii) In stating the coordinates of a point in the coordinate plane, the x - coordinate comes first, and then the y - coordinate. We place the coordinates in brackets. Hence, the coordinates of P are (4, 3) and the coordinates of Q are (– 6, – 2).

5. The Relationship between the signs of the coordinates of a point and the quadrants

   • The following relationship between the signs of the coordinates of a point and the quadrant of a point in which it lies. (i) If a point is in the 1st quadrant, then the point will be in the form (+, +), since the 1st quadrant is enclosed by the positive x - axis and the positive y - axis. (ii) If a point is in the 2nd quadrant, then the point will be in the form (–, +), since the 2nd quadrant is enclosed by the negative x - axis and the positive y - axis. (iii) If a point is in the 3rd quadrant, then the point will be in the form (–, –), since the 3rd quadrant is enclosed by the negative x - axis and the negative y - axis. (iv) If a point is in the 4th quadrant, then the point will be in the form (+, –), since the 4th quadrant is enclosed by the positive x - axis and the negative y - axis