 Hello and welcome to the session. In this session first we will discuss general equation of a line. Any equation of the form ax plus vy plus c equal to 0 where abc are the real constants with ab not 0 simultaneously. This is the general equation of a line. Next we shall discuss different forms of ax plus vy plus c equal to 0. The general equation of a line that is ax plus vy plus c equal to 0 can be reduced into various forms of the equation of a line. Consider the first form which is slope intercept form. Consider the general equation of the line that is ax plus vy plus c equal to 0. In this if we have b is not equal to 0 then this equation can be written as y is equal to minus a upon bx minus c upon b which is of the form y is equal to mx plus c. So here the slope m is equal to minus a upon b and the y intercept c is equal to minus c upon b. And if in case we have b is equal to 0 then the equation is written in the form x equal to minus c upon a which is a vertical line whose slope is undefined the x intercept is minus c upon a. So this is the slope intercept form of the equation of a line where slope is minus a upon b and the y intercept is minus c upon b. Next is the intercept form. Consider the general equation of the line ax plus vy plus c equal to 0. If we have c is not equal to 0 then this can be written as x upon a plus y upon b equal to 1 which is the intercept form of the equation of a line where we have taken this a equal to minus c upon a and b equal to minus c upon b. This is the x intercept and this is the y intercept. If we have c is equal to 0 then this equation can be written in the form ax plus vy equal to 0 which is the equation of a line passing through origin 0 intercepts on the axis. Next we shall discuss the normal form. Normal form of the general equation of the line ax plus vy plus c equal to 0 is given by x cos omega plus y sin omega is equal to p where we have cos omega is equal to plus minus a upon square root of a square plus b square sin omega is equal to plus minus b upon square root of a square plus b square then p is equal to plus minus c upon square root of a square plus b square. Proper choice of sign is made so that p should be positive. Hence this is the normal form of the equation of the line. Consider the equation root 3x plus y plus 2 equal to 0 we need to reduce this to slope intercept form. Here we have a is equal to root 3, b is equal to 1 and c is equal to 2 and the slope intercept form is given by y equal to minus a upon vx minus c upon b that is we get the slope intercept form is given by y is equal to minus a upon b x minus c upon b that is y is equal to minus root 3x minus 2. This is of the form y equal to mx plus c so from here we get m that is the slope is equal to minus root 3 and the y intercept that is c is equal to minus 2. So this is how any equation given in the general form can be reduced to slope intercept form, intercept form and normal form. Next we discuss distance of a point from a line. Consider a point p which coordinates x1, y1 and the line n is given by the equation ax plus by plus c equal to 0. We need to find the distance of the point p from the line l. Let this distance be denoted by d. So this distance d is equal to modulus ax1 plus by 1 plus c upon square root of a square plus b square. Let's try and find the distance of the point p which coordinates 4, 1 from the line given by l 3x minus 4y plus 12 equal to 0. From this point p we have x1 equal to 4, y1 equal to 1, from this line l we have a equal to 3, b equal to minus 4 and c equal to 12 and the perpendicular distance d from the point p to the line l is given by modulus ax1 plus by 1 plus c upon square root of a square plus b square. That is we get d equal to 20 upon 5 which is equal to 4. Next we have distance between two parallel lines. We know that the slope of two parallel lines are equal. So if the two parallel lines are given in the form y equal to mx plus c1 and y equal to mx plus c2 then the distance d between these two parallel lines is given by modulus c1 minus c2 upon square root of 1 plus m square and if the equation of the two parallel lines is given in general form as ax plus by plus c1 equal to 0 and ax plus by plus c2 equal to 0 then the distance between these two parallel lines given by d is equal to modulus c1 minus c2 upon square root of a square plus b square. Consider the equation of two parallel lines be given as 15x plus 8y minus 34 equal to 0 and 15x plus 8y plus 31 equal to 0. So from here we have a is equal to 15, b is equal to 8, c1 is equal to minus 34 and c2 is equal to 31. Distance d between these two parallel lines is given by modulus c1 minus c2 upon square root of a square plus b square which is equal to 65 upon 17 that is the distance between the given two parallel lines is 65 upon 17. This completes the session. Hope you have understood the general equation of a line and different forms of the general equation of the line distance of a point from a line and the distance between two parallel lines.