The position of a point in a plane is fixed w.r.t. to its distances from two axes of reference, which are drawn by the two graduated number lines XOX' and YOY', at right angles to each other at O.
The horizontal number line XOX′ is called x-axis and the vertical number line YOY′ is called y-axis. The point O, where both axes intersect each other is called the origin. The two axes together are called rectangular coordinate system.
The positive direction of x-axis is taken to the right of the origin O, OX and the negative direction is taken to the left of the origin O, i.e., the side OX′. Similarly, the portion of y-axis above the origin O, i.e., the side OY is taken as positive and the portion below the origin O, i.e., the side OY′ is taken as negative.
Coordinate of Point
The position of a point is given by two numbers, called co-ordinates which refer to the distances of the point from these two axes. By convention the first number, the x-co-ordinate (or abscissa), always indicates the distance from the y-axis and the second number, the y-coordinate (or ordinate) indicates the distance from the x-axis.
For example, the distance of the point A(3, 2) from the y-axis is 3 units and from the x-axis is 2 units.
Co-ordinates of a point P(x, y) imply that distance of P from the y-axis is x units and its distance from the x-axis is y units.
Co-ordinates of origin are (0, 0). Any point (x, 0) lies on x-axis. Any point (0, y) lies on y-axis.
(x, y) and (y, x) do not represent the same point when x ≠ y.
Quadrants
The two axes XOX′ and YOY′ divide the plane into four parts called quadrants.
Along x-axis, the positive direction is taken to the right of the origin and negative direction to its left. Along y-axis, portion above the x-axis is taken as positive and portion below the x-axis is taken as negative.
Therefore, co-ordinates of all points in the first quadrant are of the type (+, +). Any point in the second quadrant has x co-ordinate negative and y co-ordinate positive (–, +), Similarly, in the third quadrant, a point has both x and y co-ordinates negative (–,–) and in the fourth quadrant, a point has x co-ordinate positive and y co-ordinate negative (+,–).
Distance Between Two Points
Distance between two points A(x1, y1) and B(x2, y2)
$$ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$
The distance of the point (x1, y1) from the origin (0, 0) is
$$ = \sqrt{(x_1)^2 + (y_1)^2} $$
- Collinear Points: Three points A, B and C are collinear, if AB + BC = AC
- Parallelogram: If length of opposite sides are equal.
- Rectangle: If opposite sides are equal and diagonals are equal.
- Square: If all 4 sides are equal, diagonals are also equal.
- Rhombus: If all 4 sides are equal.
- Parallelogram but Not rectangle: Opposite sides are equal but diagonals are not equal.
- Rhombus but not square: All sides are equal but diagonals are not equal.
Example: Find the distance between each of the following points:
- P(6, 8) and Q(–9, –12)
- A(–6, –1) and B(–6, 11)
Example: The distance between two points (0, 0) and (x, 3) is 5. Find x.
Example: Show that the points (1, 1), (3, 0) and (–1, 2) are collinear.
Example: Find the radius of the circle whose centre is at (0, 0) and which passes through the point (–6, 8).
Section Formula
To find the co-ordinates of a point, which divides the line segment joining two points, in a given ratio internally.
Let A(x1, y1) and B(x2, y2) be the two given points and P(x, y) be a point on AB which divides it in the given ratio m : n. You have to find the co-ordinates of P.
$$ x = \frac{mx_2 + nx_1}{m + n} $$
$$ y = \frac{my_2 + ny_1}{m + n} $$
Example: Find the co-ordinates of a point which divides the line segment joining each of the following points in the given ratio:
- (2, 3) and (7, 8) in the ratio 2 : 3 internally.
- (–1, 4) and (0, –3) in the ratio 1 : 4 internally.
Midpoint Formula
The co-ordinates of the mid-point of the line segment joining two points (x1, y1) and (x2, y2) can be obtained by taking m = n in the section formula.
$$ x = \frac{x_2 + x_1}{2} $$
$$ y = \frac{y_2 + y_1}{2} $$
Example: Find the mid-point of the line segment joining two points (3, 4) and (5, 12).
Example: The co-ordinates of the mid-point of a segment are (2, 3). If co-ordinates of one of the end points of the line segment are (6, 5), find the co-ordinates of the other end point.
Centroid of Triangle
The centroid of a triangle is the point of concurrency of its medians and divides each median in the ratio of 2:1.
Let A(x1, y1), B(x2, y2) and C(x3, y3) be the vertices of the triangle . The coordiantes of the centroid are given by:
$$ x = \frac{x_1 + x_2 + x_3}{3} $$
$$ y = \frac{y_1 + y_2 + y_3}{3} $$
Example: The co-ordinates of the vertices of a triangle are (3, -1), (10, 7) and (5, 3). Find the co-ordinates of its centroid.