## Straight Lines

#### Important Formulas:

SLOPE OF LINE:

1. Let       and   be any two points.

2. The slope of the line joining  and  is defined as 

3. where   is the angle which the line makes with the positive direction of the   x-axis,   except at 

4. Which is possible only if     and the line is parallel to the y-axis.

STANDARD FORMS OF THE EQUATION OF A LINE:

5. An equation of a line  parallel to the x-axis is   and that of the x-axis itself is 

6. An equation of a line  parallel to the x-axis is   and that of the x-axis itself is 

7. An equation of a line passing through the origin and  making an angle  with the positive direction of the x-axis is 

8. An equation of a line passing through the origin having a slope   is 

9. An equation of a line passing through the origin Passing through the point   is 

10. Slope-Intercept form: An equation of a line with slope  and making an intercept  on the y-axis is 

11. Point-Slope form An equation of a line with slope  and passing through    is .

12. Two - Point form: An equation of a line passing through the points  and  

13. Intercept form: an equation of a line making intercepts   and  on the x-axis and y-axis respectively, is  

14. Parametric corm: An equation of a line passing through a fixed  point   and making an angle   with the positive direction of the x-axis is  

15. where  is the distance of an arbitrary point  on the line from the point  . Note that   and 

16. Normal form: An equation of a line such that the length of the perpendicular from the origin on it is   and the angle

17. which this perpendicular makes  with the positive direction of the x-axis is  is 

18. General form : In general , an equation of a straight line is of the form   , where  and  are real numbers,  and  both cannot be zero simultaneously.

19. The slope is  

20. The intercept on the x-axis is   and the intercept on the y-axis is  

21.   and , the positive sign being taken if  is negative and vice versa.

22. If   denotes the length of the perpendicular from  on this line, then 

23. The points   and  lie on the same side of the line if the expressions  have the same sign, and on the opposite side if they have the opposite signs.

TWO OR MORE LINES:

24. Two lines given by the equations  and   are parallel (i.e., the slopes are equal), if   

25. Two lines given by the equations  and   are  perpendicular(i.e., the product of their slopes is  -1),  if 

26. Two lines given by the equations  and   are identical if  

27. Two lines given by the equations  and   are not parallel, then  angle   between them at their point of intersection is given by    being the slopes of the two lines.

28. Two lines given by the equations  and   are not parallel, then  the coordinates of their point of intersection are 

29. Two lines given by the equations  and   are not parallel, then an equation of any line through their point of intersection is  where  is a non zero finite real number.

30. An equation of a line parallel to the line   is  and the distance between these lines is 

31. Three lines  and  are  (intersect at a point) if and only if 

32. Equations of the bisectors of the angles between two intersecting lines  and  are  

33. Equations of the lines through  and making an angle  with the line   are  where  and  where 

for_StraightLines