Two real number lines that are perpendicular to each other and that intersect at their respective zero points define a rectangular coordinate system. Often called the x-y coordinate system or x-y plane. The horizontal number line is called the x-axis and the vertical number line is called the y-axis. The point where the two axes intersect is called the origin, denoted by O. The positive half of the x-axis is to the right of the origin, and the positive half of the y-axis is above the origin. The two axes divide the plane into four regions called quadrants I, II, III, and IV, as shown in the figure below.
Each point P in the x-y plane can be identified with an ordered pair (x, y) of real numbers and is denoted by P (x, y). The first number is called the x-coordinate, and the second number is called the y-coordinate. A point with coordinates (x, y) is located lxl units to the right of the y-axis if x is positive or to the left of the y-axis if x is negative. Also, the point is located lyl units above the x-axis if y is positive or below the x-axis if y is negative. If x = 0, the point lies on the y-axis, and if y = 0, the point lies on the x-axis. The origin has coordinates (0, 0).
In the figure above, the point P (4, 2) is 4 units to the right of the y-axis and 2 units above the x-axis, and the point P’’’ (-4, -2) is 4 units to the left of the y-axis and 2 units below the x-axis.
If Two points P and Q are such that they are represented by the points (x1, y1) and (x2, y2) on the x-y plane, then the distance between two pints P and Q = ((x1 – x2)2 + (y1 – y2)2)0.5
If any point C (x, y) divides the line segment joining the points A (x1, y1) and B (x2, y2) in the ratio m: n internally,
X = (mx2 + nx1)/ (m + n)
Y = (my1 + my2)/ (m + n)
If any point C (x, y) divides the line segment joining the points A (x1, y1) and B (x2, y2) in the ratio m: n externally,
X = (mx2 – nx1)/ (m + n)
Y = (my1 – my2)/ (m + n)
Slope of a line:
The slope of a line joining two points A (x1, y1) and B (x2, y2) is denoted by m and is given by m = (y2 – y1)/ (x2 – x1) = tan , where is the angle that the line makes with the positive direction of x-axis.
Equation of line:
Slope Intercept form:
The equation of a straight line passing through the point A (x1, y1) and having slope m is given by
(y – y1) = m (x – x1)
The equation of a straight line making intercepts a and b on the axes of x and y respectively is given by
x/a + y/b = 1
Perpendicularity and parallelism:
Conditions for two lines to be parallel:
Two lines are said to be parallel if their slopes are equal.
Conditions for two lines to be perpendicular:
Two lines are said to be perpendicular if the product of slopes of two lines is equal to -1.