The co-ordinate plane is determined by two perpendicular lines, the x-axis and the y-axis. The x-axis is the horizontal axis and y-axis is the vertical axis. The point at which the two axes intersect is designated as the origin and is the point (0, 0). Any point in the plane can be expressed by a pair of ordered co-ordinates that express the location in terms of the two axes. The figure below shows the co-ordinate plane with a few points labeled with their co-ordinates.
As can be seen in the figure above, each of the points on the co-ordinate plane receives a pair of co-ordinates: (x, y). The first number x is called the x-co-ordinate and the second number y is called the y-co-ordinate. The x-co-ordinate of a point is its location along the x-axis and can be determined by the point’s distance from the y-axis (where x = 0). If the point is to the right of the y-axis, its x-co-ordinate is positive and if the point is to the left of the y-axis, its x-co-ordinate is negative. The y-co-ordinate of a point is its location along the y-axis and can be calculated as the distance from that point to the x-axis. If the point is above the x-axis, its y-co-ordinate is positive and if the point is below the x-axis, its y-co-ordinate is negative.
The co-ordinate plane is divided into four quadrants. Each quadrant is a specific region in the co-ordinate plane as shown in the figure in the following page. For example, the point (-3, 2) lies in Quadrant II, with an x-co-ordinate 3 units to the left of the y-axis and a y-co-ordinate 2 units above x-axis.
Lines and Distance
Lines and distance are fundamental to co-ordinate geometry. Some basic rules and formulae to find the distance between two points, equations of lines, angles between two lines and slope of a line are given as follows:
1. The distance between the two points (x1, y1) and (x2, y2) is given by,
2. The distance between the origin and the point (x, y) is given by,
3. If the endpoints of a line segment are (x1, y1) and (x2, y2), then the midpoint of the line segment is given by,
Lines are nothing but an infinite set of points arrayed in a straight formation. The graph of lines is represented using a linear equation. There can be one and only one line containing two distinct points from plane geometry.
Slope of a line
The slope of a line is a measurement of how steeply the line climbs or falls as it moves from left to right. For a non vertical line, it is a line’s vertical change (known as “rise”) divided by its horizontal change (known as “run”). If a line L passes through the points P(x1, y1) and Q(x2, y2), then its slope is given by,
Positive and negative slopes
We can easily determine whether the slope of a line is positive or negative just by looking at the line. If a line slopes uphill as we trace it from left to right (l3), the slope is positive. If a line slopes downhill as we trace it from left to right (l4), the slope is negative. The steeper the line, the more extreme the slope will be; the flatter the line, the smaller the slope will be. Note that an extremely positive slope is bigger than a moderately positive slope while an extremely negative slope is smaller than a moderately negative slope.
Parallel and Perpendicular Lines
Parallel lines don’t intersect. In co-ordinate geometry, they can also be described as lines with the same slope. Perpendicular lines intersect at a right angle. In co-ordinate geometry, perpendicular lines have opposite, reciprocal slopes. That is, a line with slope m is perpendicular to a line with a slope of “- (1/m)” .
Points to be noted:
1. A line that is horizontal (l1) has a slope of zero. Since there is no “rise”, y2 – y1 = 0. Thus, m = (y2 – y1)/(x2 – x1) = 0/(x2 – x1) = 0.
2. A line that is vertical (l2) has an undefined slope. Since there is no “run”, x2 – x1 = 0. Thus, m = (y2 – y1)/(x2 – x1) = (y2 – y1)/0 and we know that division by zero is not defined.
3. A line that makes an angle of 45o with the horizontal has a slope of 1 or –1. Since the “rise” equals the “run”, therefore y2 – y1 = x2 – x1 or y2 – y1 = – (x2 – x1)
4. The perpendicular distance from the point (x1, y1) to the line ax + by + c = 0 is given by,
5. The distance between two parallel straight lines ax + by + c1 = 0 and ax + by + c2 = 0 is given by,