This document presents a mathematics project on straight lines, covering formulas like distance and section formula, conditions for parallelism and perpendicularity, and equations of lines including point-slope, slope-intercept, and normal forms. It explains concepts like slopes, angles between lines, and distances between points and lines. The document includes mathematical expressions and derivations relevant to straight line geometry.

1. Kendriya Vidyalaya
Karwar
Math’s Project
2012-2013

2. Welcome to my
PowerPoint
presentation

3. Topic : Straight Line
By: Atit S Gaonkar

5. IndexA

6. EQUATIONS
A GLANCE AT X MATHS
A LINESLOPE OF
INDEXA

7. A Glance At ‘X’ Maths
• Distance Formula :
Let P(x1, y1) & Q (x2, y2)
PQ = ( [x2-x1]2 + [y2-y1]2 )

8. • Section formula :
If The Line Joining the Points P(x1, y1) & Q (x2, y2)
and in ratio m : n , the coordinates are
( [mx2+ nx1] / [m + n] , [my2+ ny1] / [m + n] )
• If m = n ; the coordinates of point are
([x2+ x1] / 2 , [y2+ y1] / 2 )

9. • The Area Of the Triangle whose vertices are
P(x1, y1) , Q (x2, y2) & R (x3, y3) equals
1/2 | x1(y2-y3) + x2(y3-y1) + x3(y1-y2) | .
• If the area equals zero then, the three points are
collinear.

10. • The Angle θ made by the line l with the positive
direction of x-axis and measured anti-clockwise, this
is called the slope or inclination.
• m = tan θ , where θ ≠ 90◦ .
Slope Of A Line

11. • Slope Of A Line When Any Two Points Of A Line
Are Given.
• ( [y2-y1] / [x2-x1] ).
• So, the slope of the line through the points P(x1, y1)
& Q (x2, y2) is given by
( [y2-y1] / [x2-x1] )

12. • Conditions For Parallelisms .
• Let we consider line l1 with it’s slope m1, and
another line l2 with it’s slope m2 .
• So, for the lines l1 & l2 to be parallel m1 should be
equal to m2 .
• i.e. m1 = m2

13. • Conditions For Perpendicularity .
• Let we consider line l1 with it’s slope m1, and
another line l2 with it’s slope m2 .
• So, for the lines l1 & l2 to be perpendicular the
products of the slopes should be equal to -1.
• i.e. m1 * m2 = -1

14. • Angle Between Two Lines.
• If we consider any line l1 passing through another
line l2 then there can be two angles :
• θ & (180 - θ ) = Φ
• tan θ = | [ m1 - m2 ] / [1 + m1m2 ] |

16. BLAH
BLAH
BLAH
BLAH
B
BLAH
BLAH
Equations of straight line

17. • If A Line Passes Through P(x1, y1) the equation of
the line is
m = ( [y-y1] / [x-x1] )
[y-y1] = m [x-x1]
1
Point
Slope Form

18. • Thus the point P(x1, y1) lies on the line with slope m
through the fixed points (x1, y1) if and only if, its
coordinates satisfy the equation
• [y-y1] = m [x-x1].

19. Two
Point Form• If a line passes from two points
P(x1, y1) & Q (x2, y2), then the equation of the line
passing through these points is
( [y-y1] / [x-x1] ) = [y2 -y1] / [x2-x1]
• ( [y-y1] ) = ( [y2 -y1] [x-x1] / [x2-x1] )
2

20. Slope
Intercept Form Suppose a line l with slope m, cuts the y-axis at a
distance ‘c’ from the origin, then ‘c’ is called the
y-intercept .
• So the equation of the line will be
[y-c] = m [x-0]
y = mx + c
3

21.  Suppose a line l with slope m, cuts the x-axis at a
distance ‘d’ from the origin, then ‘d’ is called the
x-intercept .
• So the equation of the line will be :
y = m (x – d)

22. Intercept form
• Suppose a line makes x-intercept
‘a’ and y-intercept ‘b’ on the axes, so qbviously
the line meets at (x, 0) & (0, y). So by two point
form the equation of the line is:
x / a + y / b = 1
4

23. Normal Form
• Suppose a non vertical line is known to us with the
following data :
Length of the perpendicular (normal) ‘p’ from the
origin to the line.
Angle θ which normal makes with the x-axis in
positive direction.
The slope of the line will be m = - ( cos θ ) / ( sin θ )
5

24. • So the equation of the line will be :
x cos ω + y sin ω = p.

25. BLAH
BLAH
BLAH
BLAH
B
BLAH
BLAH
General
Equations
of line

26. • The equation of first degree in two variables in the
form :
Ax + By + c = 0 , Where A, B & C are real
constants
1
The General
form

27. Slope
Intercept Form
• y = mx + c ; So in general equation form the
equation will be :
y = (-A / B ) x – ( C / B ) , where
m = (-A / B ) & c = – ( C / B )
2

28. Intercept Form
 x / a + y / b = 1, The equation in the general
form will be :
x / (-A / B) + y / (-C / B) = 1, where
x-intercept is (-A / B) & y-intercept is (-C / B)
3

29. Normal form
• x cos ω + y sin ω = p , the equation in the
general form will be :
A/( A2 + B2 )1/2 x + B/( A2 + B2 ) 1/2 y = C/( A2 + B2 )
where cos ω = ± A/( A2 + B2 )1/2
sin ω = ± B/( A2 + B2 ) 1/2
c = ± C/( A2 + B2 ) 1/2
4

30. • The distance of a line Ax + By + C = 0
perpendicular to the point P(x1, y1) is given by ‘d’.
d = | (Ax1 + By1 + C ) / (A2 + B2 ) 1/2 |
Distance Of A Point
From A Line

31. • Let two lines be :
y = mx + c1 .
y = mx + c2 .
Then the perpendicular distance between the two
line is given by ‘d’.
d = | c1 – c2 | / ( 1 + m2 ) 1/2
Distance Between
two parallel Line

32. • Let two lines be :
Ax + By + C1 = 0
Ax + By + C2 = 0
Then the perpendicular distance between the two
line is given by ‘d’.
d = | C1 – C2 | / ( A2 + B2 ) 1/2