1. Introduction

Coordinate Geometry is the branch of mathematics where we study the position of points on a plane using numbers called coordinates.

The plane on which we locate points is called the Cartesian Plane or Coordinate Plane.

2. Cartesian System

A Cartesian Plane consists of two mutually perpendicular lines:

• X-axis → The horizontal line
• Y-axis → The vertical line

The point where these two axes intersect is called the Origin (O).

Coordinates of the origin are:
(0,0)(0, 0)

3. Coordinates of a Point

Every point on the plane is represented by an ordered pair (x,y)(x, y), where:

• xxx: Abscissa → Distance from the Y-axis
• yyy: Ordinate → Distance from the X-axis

For example, P(x,y)P(x, y) represents a point on the plane.

4. Quadrants

The X and Y axes divide the plane into four quadrants:

5. Distance Formula

The distance between two points A(x1,y1)A(x_1, y_1) and B(x2,y2)B(x_2, y_2) is:

AB=(x2−x1)2+(y2−y1)2AB = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}

6. Section Formula

The coordinates of a point P(x,y)P(x, y) dividing the line joining A(x1,y1)A(x_1, y_1) and B(x2,y2)B(x_2, y_2) internally in the ratio m:nm:n are:

P(mx2+nx1m+n,my2+ny1m+n)P\left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right)

If it divides externally, then:

P(mx2−nx1m−n,my2−ny1m−n)P\left( \frac{mx_2 – nx_1}{m – n}, \frac{my_2 – ny_1}{m – n} \right)

7. Midpoint Formula

The midpoint MMM of the line joining A(x1,y1)A(x_1, y_1)A(x1​,y1​) and B(x2,y2)B(x_2, y_2)B(x2​,y2​) is:

M(x1+x22,y1+y22)M\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

8. Area of a Triangle

If vertices of a triangle are A(x1,y1)A(x_1, y_1)A(x1​,y1​), B(x2,y2)B(x_2, y_2)B(x2​,y2​), and C(x3,y3)C(x_3, y_3)C(x3​,y3​), its area is:

Area=12[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)]\text{Area} = \frac{1}{2} [x_1(y_2 – y_3) + x_2(y_3 – y_1) + x_3(y_1 – y_2)]

9. Collinearity of Points

Points A(x1,y1)A(x_1, y_1), B(x2,y2)B(x_2, y_2), and C(x3,y3)C(x_3, y_3) are collinear if the area of the triangle formed by them is zero:

12[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)]=0

10. Important Points

• Points on the Y-axis have coordinates (0,y)(0, y).
• Points on the X-axis have coordinates (x,0)(x, 0).
• The origin is at (0,0)(0, 0).

11. Summary Table