CLASH 9TH NCERT MATH EXAMPLER EXERCISE 3 QUESTION NUMBER 3 AND 4 COORDINATE GEOMETRY

Coordinate Geometry and Cartesian System Detailed Description Coordinate Geometry is an important branch of mathematics that combines algebra and geometry. It helps us understand geometrical concepts using numbers and positions. Instead of only drawing shapes, coordinate geometry allows us to locate exact positions of points on a plane using numerical values. This method makes geometry more logical, accurate, and useful in real life. The study of coordinate geometry is very important for students of Class 9 because it forms the base for higher mathematics. Many topics in Class 10 and above such as graphs of equations, linear equations, trigonometry, and analytical geometry depend on the concepts of coordinate geometry. Learning this chapter properly helps students build strong mathematical thinking and problem solving skills. Meaning of Coordinate Geometry Coordinate Geometry is the study of geometry using a coordinate system. In this system, the position of a point is described by numbers instead of words. These numbers tell us how far the point is from fixed reference lines. Because of this, coordinate geometry is also known as analytical geometry. This system was developed by the mathematician Rene Descartes. Due to his contribution, the coordinate system used in geometry is called the Cartesian System. The Cartesian system helps us convert geometric problems into algebraic problems, making them easier to solve. Cartesian System The Cartesian System is the basic structure of coordinate geometry. It consists of two number lines that are perpendicular to each other and intersect at a common point. These two lines divide the plane into different regions and help us locate points accurately. One line is horizontal and is called the X axis. The other line is vertical and is called the Y axis. The point where both lines intersect is called the origin. The origin represents zero on both number lines. The Cartesian plane is the flat surface formed by the X axis and Y axis together. Every point on this plane has a unique position that can be described using two numbers. Coordinates of a Point The position of a point in the Cartesian plane is represented by two numbers written in a specific order. These numbers are called the coordinates of the point. The first number tells the horizontal distance of the point from the Y axis. This number is called the x coordinate. The second number tells the vertical distance of the point from the X axis. This number is called the y coordinate. Together, the x coordinate and y coordinate describe the exact position of a point on the plane. By using coordinates, we can easily plot points and study shapes. Axes in the Cartesian Plane The X axis and Y axis play a very important role in coordinate geometry. The X axis is the horizontal line. All points lying on the X axis have a y coordinate equal to zero. Points on the right side of the origin have positive x values, while points on the left side of the origin have negative x values. The Y axis is the vertical line. All points lying on the Y axis have an x coordinate equal to zero. Points above the origin have positive y values, while points below the origin have negative y values. Both axes together help divide the plane into different regions. Quadrants of the Cartesian Plane The X axis and Y axis divide the Cartesian plane into four regions called quadrants. Each quadrant has a specific sign for x and y coordinates. First quadrant In this quadrant, both x and y values are positive. Points in this region lie to the right of the Y axis and above the X axis. Second quadrant In this quadrant, x values are negative and y values are positive. Points lie to the left of the Y axis and above the X axis. Third quadrant In this quadrant, both x and y values are negative. Points lie to the left of the Y axis and below the X axis. Fourth quadrant In this quadrant, x values are positive and y values are negative. Points lie to the right of the Y axis and below the X axis. Points that lie on the X axis or Y axis do not belong to any quadrant. Plotting Points on the Cartesian Plane Plotting points is one of the most important skills in coordinate geometry. It helps students visually understand positions and shapes. To plot a point on the Cartesian plane, we first draw the X axis and Y axis. We mark equal units on both axes. Starting from the origin, we move horizontally according to the x coordinate and then move vertically according to the y coordinate. After reaching the correct position, we mark the point clearly and label it.