Line :-
A line is one dimensional figure which has length but no breath.
FIG. RayOB & RayOA joined together.
It is also defined as a combination of two rays
joined together at a common vertex but in
opposite direction.
Both the side it can be extended infinitely.
There are are different types of lines like straight line, vertical line, thick line , thin line, horizontal line , zigzag line, curved line , spiral line ,diagonal line , skew line ,slant/oblique line etc.
EQUATION OF LINE
There are two form of equation of line ...
1) ax+by=0 (Line passes through origin)
2) ax+by+c=0 (Line does not pass through origin)
TYPES OF EQUATION OF LINE : -
1) TWO POINT FORM :
When Two points A(x₁,y₁) & B(x₂, y₂)
of a line is known then the equation of line
is given by ...
y-y₁ y₁- y₂
------ = --------
x-x₁ x₁-x₂
2) Slope Point form :
When a point A(x₁,y₁) & slope (m)
of a line is known then the equation
of a line is given by ..
y-y₁=m(x-x₁)
3) Double Intercept form :
When two intercept i.e
x-intercept= a
& y-intercept= b
of a line is known then the equation
of a line is given by
x/a +y/b =1
4) SLOPE - INTERCEPT FORM :
When slope (m) & Y-intercept-c
of a line is known then the equation
of a line is given by
y=mx+c
EQUATIONS OF AXES
1) Equation of x-axis is y=0
2) Equation of y-axis is x=0
Note:
1)If a line is perpendicular to x-axis then
it's equation is given by x= ± a
2) If a line is perpendicular to y-axis then
it's equation is given by y= ± b
TO FIND SLOPE :
To find slope of a line there are three formulae
1) If Ө is the angle made by the line with the + direction of x-axis then the slope of the line is given by ...
slope = TanӨ
2) When A(x₁,y₁) & B(x₂, y₂) two points of two line s are known then slope is given by
y₂-y₁
slope = -----------
x₂-x₁
3)When ax+by+c=0 is the equation of a given line then the slope is given by
slope = -a / b
TO FIND SLOPE OF AXES
1) Slope of x-axis = 0
2) Slope of y-axis = not defined =∞
* * IF m₁ & m₂ slopes of two lines are then
1) If two lines are parallel then
m₁ = m₂
2) If two lines are perpendicular then
m₁.m₂= -1
3) If two lines are intersecting then angle between them is given by
TanӨ= | m₁ - m₂/ 1 - m₁. m₂|
* If a point is on x- axis then it's
coordinate is given by
(± a , 0)
* If a point is on y-axis then it's
coordinate is given by
( 0, ± b)
*If u=a₁x+b₁y+c₁=0 &
v=a₂x+b₂y+c₂=0
are two lines then equation of a
line passing through the
intersection of these two line is given by
u+kv=0
*Joint equation of two line is given by
u.v=0
* Joint Equation / Homogenous equation
of pair of line passing through origin is
given by
ax²+2hxy+by²=0
* General equation of pair of line is given by
ax²+2hxy+by²+2gx+2fy+c=0
* If the two line are perpendicular then
a+b=0
*If two line are parallel then
h²-ab=0
* If the two lines are intersecting then
the angle between them is given by
TanӨ= | 2√ h²-ab/a+b |
**Shortest Distance between a Point P(x₁,y₁)
& a point M on the line ax+by+c=0 is
given by
PM = | ax₁+by₁+c₁/√ a²+b²|
***
In 3-Dimension :
If a line makes an angle 𝝰 ,𝞫 &𝜸
with the + direction of x-axis,y-axis
&z-axis respectively.
Then cos 𝝰 ,cos 𝞫 , & cos𝜸 are
known as Direction cosine of the line.
It is also known as dc's.
also cos𝝰=l ,cos𝞫=m & cos𝜸=n
**cos²𝝰 +cos²𝞫 +cos²𝜸 = 1
i.e. l²+m²+n² =1
* If a/cos𝝰=b/cos𝞫=c/cos𝜸 then a, b, c
are known as direction ratios.
It's is denoted by dr's.
*If A(x₁,y₁,z₁) & B(x₂, y₂,z₂) are two point
then it's direction ratio is given by
a=x₂-x₁ , b=y₂-y₁,c =z₂-z₁
*The Vector equation of the line passing through A(ā)& parallel to Б is given by
r=ā+⋋b
* Cartesian equations of the line passing through A(x₁,y₁,z₁) & having direction
ratio a, b & c are
x-x₁/a = y-y₁/b =z-z₁/c
