 Hello and welcome to the session. In this session we will discuss how to determine the equation of the line given a set of geometric conditions pertaining to the line. Now there are basically two types of problems in coordinate geometry. One is that where we are given algebraic equation and we draw its graph. Another is where we are given a set of conditions pertaining to the figure and we have to find its algebraic equation. Here we will discuss the second problem regarding the lines. As we deal with straight lines we often refer to steepness or slant of a line. Now slope is used to measure the steepness of a line. Slope is the ratio of vertical change of distance to the horizontal change of distance when we move from one point on the line to another. Look at this graph. As we move from point P1 with coordinates x1, y1 to point P2 with coordinates x2, y2. We have vertical change which is given by y2 minus y1 and horizontal change which is given by x2 minus x1. So slope of this line will be given by vertical change upon horizontal change and this is equal to y2 minus y1 whole upon x2 minus x1. So it is clear that if a with coordinates x1, y1 and b with coordinates x2, y2 be any two points on the line then slope of the line which is denoted by m is given by y2 minus y1 whole upon x2 minus x1 and x2 is not equal to x1. Now let us discuss conditions for parallel and perpendicular lines. If two lines have slopes m1 and m2 then these two lines are parallel if and only if m1 is equal to m2 and these two lines are perpendicular if and only if m1 into m2 is equal to minus 1. Now let us discuss x-intercept and y-intercept. If a line needs the x-axis at point A then the distance of point A from origin O that is OA is called x-intercept. Its corresponding point on x-axis is given by the ordered pair x0 and if a line needs the y-axis at point B then the distance of point B from origin O that is OB is called y-intercept. Its corresponding point on y-axis is given by the ordered pair 0y. Now let us discuss equation of line. We have various forms of the equations of straight lines. The first form is given by slope-intercept form. This form is used when slope and intercept on the y-axis are given let y-intercept be equal to c that is the point with coordinates 0c. Let slope be equal to m and point P with coordinates xy be any point on the line then slope of the line will be given by m is equal to y2 minus y1. So here it will be y minus c whole upon x minus 0 which implies that m is equal to y minus c whole upon x. This further implies m into x that is mx is equal to y minus c which implies y is equal to m into x that is mx plus c which is the required equation of the line in slope-intercept form. Second we have point slope form. This form is used when slope of the line and a point on it are given. Suppose we are given a line whose slope is given by m and it is passing through the point with coordinates x1 y1 then we consider another point with coordinates xy on the line. Now we know that its slope m will be given by y minus y1 whole upon x minus x1 which implies that y minus y1 is equal to m into x minus x1 the whole. This is the required equation of the line in point slope form. Now let us discuss some examples. Here we have our first example. Find equation of the line which passes through the point with coordinates 1 3 and whose y-intercept is 2. Now we start with the solution of this question. Now here we are given y-intercept is equal to 2 that is c is equal to 2. We know that equation of the line in slope-intercept form is given by y is equal to mx plus c which implies that y will be equal to mx plus 2 as the value of c is given as 2. It is also given that line passes through the point with coordinates 1 3 so we put the value of x as 1 and the value of y as 3 in this equation. So we get 3 is equal to m into 1 plus 2 which implies that 3 is equal to m plus 2 which implies m is equal to 3 minus 2 which is equal to 1. Now we have got the value of m as 1. So the required equation of the line is given by y is equal to 1 into x plus 2 which implies that y is equal to x plus 2. Let us consider another example. Find the equation of the line passing through the point with coordinates 2 minus 1 and parallel to the line 2x minus y is equal to 4. Let us start with the solution of the given problem. Here we are given the equation 2x minus y is equal to 4. Let us write this equation in y is equal to mx plus c form. It will be y is equal to 2x minus 4. On comparing these two equations we get the value of m as 2 so we get slope is equal to 2. Now the given line is parallel to the line passing through the point with coordinates 2 minus 1 and we know that two lines are parallel if and only if their slopes are equal. So we can say that slope of the given line which is equal to 2 will be equal to slope of the line passing through the point with coordinates 2 minus 1 for the line passing through the point with coordinates 2 minus 1 will be equal to the ordered pair x1 y1 and slope m will be equal to 2. So its equation is given by y minus y1 is equal to m into x minus x1 the whole that is y minus of minus 1 will be equal to 2 into x minus 2 the whole. This implies that y plus 1 is equal to 2 into x that is 2x minus 2 into 2 that is 4. This further implies y is equal to 2x minus 4 minus 1 which implies y is equal to 2x minus 5. So this is the equation of the line passing through the point with coordinates 2 minus 1 and parallel to the line 2x minus y is equal to 4. Thus in this session we have discussed how to determine the equation of the line given a set of geometric conditions pertaining to the line. This completes our session. Hope you enjoyed this session.