 Hello and welcome to the session. In this session first we will discuss slope of a line. A line in a coordinate plane forms two angles with the x-axis which are supplementary. This angle theta made by the line L with the positive direction of the x-axis is called the inclination of the line. So we say if theta is the inclination of a line L then tan theta is called the slope or gradient of a line L. Slope of a line is denoted by m that is we have m is equal to tan theta and theta is not equal to 90 degrees. Slope of a horizontal line is 0 because for a horizontal line the inclination that is theta would be 0 degrees and so we get slope m is 0 in this case. Slope of a vertical line is undefined because for vertical line theta that inclination is 90 degrees and so m equal to tan 90 is undefined. Suppose that the inclination of a line is given as theta equal to 45 degrees so the slope m would be equal to tan 45 degrees that is 1. So slope of a line with inclination 45 degrees is 1. Next we shall discuss slope of a line when coordinates of any two points on the line are given. Let p and q be any two points on this non-vertical line L with inclination theta let the coordinates of the point p be x1, y1 and coordinates of the point q be x2, y2 and this x1 is not equal to x2 otherwise the line will become perpendicular to x axis and its slope will not be defined and inclination of the line L may be acute or obtuse. So the slope m of a non-vertical line passing through the points p and q is given by y2 minus y1 upon x2 minus x1 and here we have x1 is not equal to x2. Consider a point p with coordinates minus 2, 3 and a point q with coordinates 8 minus 5. So the slope of the line passing through the points p and q is given by m equal to y2 that is minus 5 minus y1 that is 3 upon x2 that is 8 minus x1 that is minus 2 and this comes out to be equal to minus 4 upon 5. Next we shall discuss conditions for parallelism and perpendicularity of lines in terms of their slopes. Consider a line L1 in the coordinate plane with slope m1 line L2 in the coordinate plane with slope m2 2 non-vertical lines L1 and L2 are parallel if and only if their slopes are equal that is the line L1 would be parallel to the line L2 if we have their slopes m1 is equal to m2. 2 non-vertical lines L1 and L2 are perpendicular to each other if and only if their slopes are negative reciprocals of each other. The slope of the second line m2 is equal to minus 1 upon m1 or we can also write it as m1 m2 is equal to minus 1. Next we shall discuss angle between two lines. Consider a line L1 with slope m1 line L2 with slope m2 and suppose that theta is the acute angle between the lines L1 and L2 then we have tan theta is equal to modulus m2 minus m1 upon 1 plus m1 m2 where m1 minus m2 is equal to minus 1 upon m2 minus m1 plus m1 m2 is not equal to 0 and if we have some obtuse angle phi then this phi would be equal to 180 degrees minus theta so using this we can find the obtuse angle phi between the two lines L1 and L2. Suppose slope of one line is given by m1 equal to 1 upon 2 slope of the other line given by m2 is equal to 3 and suppose that theta is the angle between the two lines then tan theta is equal to modulus m2 minus m1 that is 3 minus 1 upon 2 upon 1 plus m1 into m2 which is equal to modulus 5 upon 2 upon 5 upon 2 that is equal to modulus 1 which is 1. So we have tan theta equal to 1 which implies that theta is equal to 45 degrees. So the angle between two lines is 45 degrees. Next we shall discuss collinearity of three points. Three points given by ABC are collinear if and only if slope of ABC is equal to slope of BC. Consider a point A which coordinates 5 to the power 5 to the power 5 1 B which coordinates 1 minus 1 and C which coordinates 11 4. Let's see if these three points are collinear or not. Now slope of AB is equal to y2 minus y1 that is minus 1 minus 1 upon x2 minus x1 which comes out to be equal to 1 upon 2 then the slope of ABC. Next we find slope of BC this is equal to y2 minus y1 upon x2 minus x1 which comes out to be equal to 1 upon 2. So you can see that slope of AB is equal to slope of BC and thus we can say that the points ABC are collinear points. This completes the session. Hope you have understood the concept of slope of a line.