Coordinate geometry

In this section, we will learn the basics of coordinate geometry. Coordinate geometry is a branch of mathematics  that helps us to understand geometry and location using coordinate points. 

  1. Cartesian plane: A plane in which two lines are placed perpendicular to each other and the position of points on the plane are located by referring to these lines. It is also called the coordinate plane or x-y plane.
    i) The lines are called coordinate axes.
    ii) The horizontal line is called the x-axis.
    iii) The vertical line is called the y-axis.

  2. Origin: The point at which both coordinate axes intersect is called the origin.

  3. Quadrants: The coordinate axes divide the plane into four parts. These four parts are called quadrant numbered anticlockwise I, II, III and IV.

  4. x- coordinate or abscissa: It is the perpendicular distance of a point from the y-axis measured along the x-axis.

  5. y-coordinate or ordinate: It is the perpendicular distance of a point from the x-axis measured along the y-axis.

  6. If x and y are the abscissa and ordinate of point P respectively, then (x, y) are called the coordinates of point P. Also written as P(x, y).

  7. The coordinates of any point on the x-axis are of the form (x, 0).

  8. The coordinates of any point on the y-axis are of the form (y, 0).

  9. The coordinates of the origin are (0, 0).

  10. Signs of coordinates in various quadrants –
    Quadrant x y
    I + +
    II +
    III
    IV +
    coordinate geometry

Coordinate geometry formulas

  1. DISTANCE FORMULA
    i) The distance between two points P (x_{1},y_{1}) and Q (x_{2},y_{2}) can be given by

    Distance formula
    PQ = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}

    This is called the distance formula.

    ii) Distance of a point P(x, y) from origin O (0,0) is given by

    OP =\sqrt{x^{2}+y^{2}}
  2. SECTION FORMULA:
    i) The coordinates of point P(x, y) which divides the line segment joining the points A (x_{1},y_{1}) and B(x_{2},y_{2}) internally in the ratio m_{1}:m_{2} are –

    Section formula
    P(x, y) =\frac{m_{1}x_{2}+m_{2}x_{1}}{m_{1}+m_{2}} , \frac{m_{1}y_{2}+m_{2}y_{1}}{m_{1}+m_{2}}


    ii) If P(x, y) divides the line segment joining the points A (x_{1},y_{1}) and B (x_{2},y_{2}) internally in the ratio k:1 , then coordinates of point P will be,

    P(x, y) = \frac{kx_{2}+x_{1}}{k+1}, \frac{ky_{2}+y_{1}}{k+1}

    iii) ) If P(x, y) is the mid-point of the line segment joining the points A(x_{1},y_{1}) and B(x_{2},y_{2}) then it divides the line segment in the ratio 1:1  , then coordinates of point P will be,

    P(x, y) = \frac{x_{1}+x_{2}}{2} , \frac{y_{1}+y_{2}}{2}

  3. Area of triangle

In triangle ABC, whose vertices are A (x_{1},y_{1}), B (x_{2},y_{2}) and C (x_{3},y_{3}) then the area of the triangle is given by ,

Area of triangle
Area of triangle ABC = \frac{1}{2}[x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2})]

 If area of the triangle is zero, then the three points A, B, and C lie on a line i.e. they are collinear. 

These were some basic concepts and formulas. Further we will look at coordinate geometry of straight lines.

STRAIGHT LINES FORMULAS