Analytic Geometry

  • Introduction to Analytic Geometry

    Geometry is the one of the most ancient branches of mathematics, concerned with the properties of space and object – points, lines, angles, planes, surfaces and solids in space. Generalization of geometry known as “Analytic Geometry” was in vented and first used in 1637 by a French Mathematician and philosopher, Rene Descarte (1596–1650 A.D). He […]

  • Coordinate System

    Cartesian coordinates are defined through the use of two coordinate lines, one horizontal and the other vertical. Let their point of intersection be , to which we call the origin and the real number of both lines is represented by . The two lines are called the coordinate axes. The horizontal line  is called the X–Axis and the vertical line  is called the Y–Axis. A point […]

  • Distance Formula

    Let and are any two points on the line. We find the distance between P and Q. For This draw PM and QN perpendicular to X–Axis. From P draw PR perpendicular to QN. Consider the right angle triangle I. First we find the distance between P and R. II. Secondly, we find the distance between […]

  • Ratio Formula and Mid Point Formula

    Let and be any two points on the line. Let a point be the point which divides PQ in the ratio i.e. PR : PQ = From P, R and Q draw PM, RN and OL perpendicular to x-axis. From ‘P’ draw to RN. From ‘R’ draw to QL. Since the right triangles and and […]

  • Translation of Axes

    If in the plane with given and axes, new coordinate axes are chosen parallel to the given ones, we say that there has been a translation of axes in the plane. Let be any point in -plane. Let be the fixed point in the - plane. We draw two perpendicular axes through , -axis is […]

  • Rotation of Axes

    Let the -coordinate system be rotated through an angle , such that the range of the angle is about the same origin . The new coordinate system is -coordinate system as shown in the given diagram. Since triangle is a right triangle with so, Since are the coordinates of the point with respect to new […]

  • Slope of a Line

    Inclination of a Line: Angle from X-axis to any given non horizontal line is called inclination of line . Here is the inclination where , measured in the counter clockwise direction from the positive X-axis to the line . Slope of a Line: If is the inclination of a non-vertical straight line , then its […]

  • Slope of a Line through Two Points

    Let and be any two points on the given line . Also consider be the inclination of the line as shown in the given diagram. From point draw perpendicular on X-axis also from draw perpendicular on X-axis. Now from the given diagram, consider the triangle , from the definition of slope we take Now by […]

  • X and Y Intercepts of a Line

    When represents the straight line graphically the two main attributes of will come out, one is the -intercept and the other is -intercept of the straight line. These two concepts are very simple and easy to understand when draw straight line graphically. -intercept is defined as when we draw a straight in a Cartesian plane […]

  • Medians of a Triangle are Concurrent

    The medians of any triangle are concurrent and that the point of concurrency divides each one of them in the ratio 2:1. Consider the triangle as shown in the diagram and suppose that , and be the vertices of the given triangle . As we know that median is defined as the line segment joining […]

  • Slope Intercept Form of Equation of Straight Line

    Consider the straight line . Let be any point on the given line . Suppose that be the inclination of the line as shown in the given diagram, i.e. Take as a -intercept of the straight line because it cutting the -axis at the point , i.e. -intercept. From point draw perpendicular on -axis also […]

  • Equation of a Line with X Intercept

    Consider the straight line and be the inclination of the straight line as shown in the given diagram now the slope of the represented by . Let be any point on the given line . Let be the X-intercept of the straight line, so the line must passes through the point . Take as X-intercept […]