Guys ever got confused with all the equations of straight lines including different forms??
Well, here goes a simple way to understand and remember the fundamentals of straight lines.
.
is normal form with
Well, here goes a simple way to understand and remember the fundamentals of straight lines.
Definition
of a Straight Line:
A straight line is a curve such every point on the
line segment pointing any two points on it lies on it.
** Note:
1) Every
first degree equation in x, y represents a straight line.
When
we say that a first degree equation in x, y i.e., represents a line ax+by+c=0, it means means that all points (x, y) satisfying ax+by+c=0 lie along the
line. Thus, a line is also defined as the locus of a point satisfying condition ax+by+c=0 where a, b, c
are constants.
**
Slope (Gradient) of a line:
The trigonometric tangent of the angle that a line
makes with the positive direction of the x-axis in anti-clock wise sense is
called the slope (or) gradient of the line.
Slope of line is generally denoted by ‘m’ thus
Note:
1) A
line parallel to x-axis makes angle 0 degrees.
2) Perpendicular
to x-axis makes an angle 90 degrees with x-axis
3) The
angle of inclination of a line with positive direction of x-axis in
anti-clockwise sense always lies between 0 degrees and 180 degrees.
4) If m>0 then
is acute
is acute
If m<0 then is obtuse.
is obtuse.
** Slope of a
line in terms of coordinates of any two points on it:
Let
and
be two points
on a line making an angle with the
positive direction of x-axis then draw PL, QM perpendicular to x-axis and PN is
perpendicular on QM.
andbe two points
on a line making an angle with the
positive direction of x-axis then draw PL, QM perpendicular to x-axis and PN is
perpendicular on QM.
**** Angle
between two lines:
The
angle between lines
having slopes m1 and m2 is given by
between lines
having slopes m1 and m2 is given by
*** Intercepts
of a line on the axes:
If a straight line cuts x-axis at A and y-axis at B.
then OA and OB are known as the intercepts of the line on x-axis and y-axis
respectively.
**** Different
forms of the equation of a straight line
**
Slope intercept form of a line:
The equation of line with slope m and making an intercept 'c' on y-axis is y = mx+c
Note:
If line passes through origin, c= 0
Therefore, the equation of a line passing through origin
is y = mx.
***Reduction
of general form to slope intercept form:
The general form of the equation of a line is Ax+By+C = 0
**** The
point slope form of a line:
The equation of a line which passes through point (x1, y1)and has slope m
is y-y1 = m(x-x1)
**** Two
point form of a line:
The equation of line passing through two points A(x1, y1) , B(x2, y2) is 
** Note:
The equation of a line passing through two points (x1, y1) and (x2, y2) can also in
determinant form
*** Intercept
form of a line:
The equation of line which cuts off intercepts a and
b respectively from x and y –axes is 
***
Reduction of general equation of line to intercept form:
The general equation of line is Ax+By+C = 0
Ax+By = -C
Intercept on x-axis
.
Intercept on y-axis
*** Normal or
perpendicular form of a line:
The equation of straight line upon which the length
of perpendicular from origin is P and this perpendicular makes an angle
with x-axis is 
.
with x-axis is .
Reduction
of general form of line to the normal form:
Ax+By+C = 0 is
is normal form with
**** Straight
line: Symmetric form: (or) Distance form of a line:
The equation of line passing through (x1, y1) and making
angle with positive
direction of x-axis is 
Where 'r' is distance of
point (x, y) on the line from the point (x1, y1)
**** Note:
Co-ordinates of any point on the line at a distance 'r' from the given
point (x1, y1) is
(or)
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