A straight line is the shortest distance between two points. It can be defined using different forms of equations like slope-intercept, two point, and normal forms. The slope of a line is calculated as the tangent of the angle between the line and the x-axis. Parallel lines have equal slopes, while perpendicular lines have slopes that multiply to -1. A line's equation can be found using various properties like slope, two points on the line, or intercepts with the axes.

1. STRAIGHT LINE
A straight line is the minimum distance
between two points.

2. • Equation of the x-axis is y=0
• Equation of the y-axis is x=0
• The equation of a line parallel to y axis at a distance
‘a’ from x axis is x=a
• The equation of a line parallel to x axis at a distance
‘b’ from y axis is y=b

3. SLOPE(Gradient)
• The slope of the line is the tangent of the
angle made by the line with positive direction
of X – axis measured in the anticlockwise
direction.
• slope = m = tan 

4. • Obtuse angle-slope is negative
• Acute angle-Slope is positive

5. • The slope of any line parallel to X axis is zero.
• Slope of any line parallel to Y axis is infinity
• The slope of the line joining two points (x1,y1)
and (x2,y2) is
m = tan  =
1
2
1
2
x
x
y
y



6. • When two or more lines are parallel then their
slopes are equal
• When two lines are perpendicular then the
product of their slopes is –1
i.e., m1m2 = -1

7. Different Forms of Equation of Straight
Line
• Slope-Intercept form
• Slope –One point form
• Two Points form
• Intercept form
• Normal form
• Parametric form.

8. Slope-Intercept form
y = mx + c
o
D
M
A
y -c
B (O, C)
X
XI
Y

9. • Any line passing through the origin does not
cut y – axis (c = 0) i.e., y – intercept is zero.
Therefore its equation is y = mx
• Any line which is parallel to x – axis has slope
equal to zero. (Because m = 0)
Therefore its equation is y = c

10. Example:
1. Find the equation of a straight line whose
(i) Slope is four and y intercept is –3
(ii) Inclination is 300 and y intercept is 5

11. Example
Calculate the slope and y intercept of the line
2x – 3y + 1= 0
Ans:Slope = ; y intercept =
3
2
3
1

12. Slope one point form
y – y1 = m (x – x1).

13. Example
Find the equation of a straight line passing
through (-4,5) and having slope –
Ans: 2x + 3y – 7 = 0
3
2

14. Two Points Form
1
2
1
1
2
1
x
x
x
x
y
y
y
y






15. Example
Find the equation of the straight line passing
through the points (3,6) and (-2,5).
Ans: x – 5y +27 = 0

16. Intercept form
1


b
y
a
x

17. Example
Find the intercepts cut off by the line
2x – 3y + 5 = 0 on the axes.
Ans: The x – intercept =
The y – intercept=
2
5

3
5

18. Note
General equation of a straight line as
ax + by + c = 0
 by = - ax – c
i.e. y =
Now, comparing this with the equation y = mx + c,
we get
slope = m = =
b
c
b
ax


b
a








y
of
t
coefficien
x
of
t
coefficien

19. Normal Form

20. Example
Ans:

21. Distance form (or) Symmetric form
(or) Parametric form

22. Example
Determine the equation of a line passes through
the point (–1, – 2) and makes an angle of 30o
with the positive direction of x-axis, in
parametric form. Find the coordinates of a
point at distance of 2 units.
Answer:
The coordinates of the point are

23. Note

24. Example
Find the length of the perpendicular from (3,2)
on the line 3x+2y+1=0
Ans:
13
14

25. Family of Straight lines
• Intersection of two straight lines
The point of intersection of two straight
lines is obtained by solving their equations.

26. • Concurrent lines
Three or more straight lines are said to be
concurrent when they all pass through the
same point. That point is known as point of
concurrency.

27. Example
Show that the lines 3x+4y = 13, 2x-7y+1 = 0 and
5x-y=14 are concurrent.

28. Equation of Straight lines passing through the
intersection of two given straight lines

29. Angle between two straight lines

30. Example
Find the angle between the straight lines
3x-2y+9=0 and 2x+y-9=0
Answer:

31. Angle Bisector
• The angle bisector has equal perpendicular
distance from two lines.

32. Example
Find the equations of the bisectors of the angles
between the straight lines 3x + 4y + 3 = 0 and
4x+ 3y +1 = 0.
Answer : x-y-2=0 and 7x+7y+4=0

33. Algorithm to find the bisectors of acute
and obtuse angles between two lines
• Let the equations of the two lines be
a1x + b1y + c1 = 0 and
a2x + b2y + c2= 0.
To find the bisectors of the obtuse and acute angles
between the lines we proceed as follows:
• Step I: First check whether the constant terms
c1and c2 in the two equations are positive or not.
Suppose not, then multiply both the sides of the
given equations by -1 to make the constant terms
positive.

34. • Step II: Determine the symbols of the
expression a1a2 + b1b2
• Step III:
• If a1a2 + b1b2> 0, then the bisector
corresponding to “ + “ symbol gives the
obtuse angle bisector and the bisector
corresponding to “ - “ is the bisector of the
acute angle between the lines .

35. • If a1a2 + b1b2< 0, then the bisector
corresponding to “ + “ and “ - “ symbol give
the acute and obtuse angle bisectors
respectively.

36. Example
Find the equation of the acute angle bisector of
lines 4x - 3y + 10 = 0 and -6x + 8y - 5 = 0.
Answer: 2x+2y+15=0