Straight Line
A straight line is the locus of all those points which are collinear with
two given points.
General equation of a line is ax + by + c = 0
Note
l We can have one and only one line through a fixed point in a given direction.
l We can have infinitely many lines through a given point.
Slope (Gradient) of a Line
The trigonometrical tangent of the angle that a line makes with the
positive direction of the X-axis in anti-clockwise sense is called the
slope or gradient of the line.
So, slope of a line, m = tan θ
where, θ is the angle made by the line with positive direction of X-axis.
Important Results on Slope of Line
(i) Slope of a line parallel to X-axis, m = 0.
(ii) Slope of a line parallel to Y-axis, m = ∞.
(iii) Slope of a line equally inclined with axes is 1 or −1as it makes an angle of
45° or 135°, with X-axis.
(iv) Slope of a line passing through ( x1 , y1) and ( x2 , y2 ) is given by
y −y
m = tanθ = 2 1 .
x2 − x1
Angle between Two Lines
The angle θ between two lines having slopes m1 and m2, is
m − m1
tanθ = 2 .
1 + m1m2
, (i) Two lines are parallel, iff m1 = m2.
(ii) Two lines are perpendicular to each other, iff m1m2 = − 1.
Equation of a Straight Line
General equation of a straight line is Ax + By + C = 0.
(i) The equation of a line parallel to X-axis at a distance b from it, is
given by
y=b
(ii) The equation of a line parallel to Y-axis at a distance a from it, is
given by
x=a
(iii) Equation of X-axis is
y=0
(iv) Equation of Y-axis is
x=0
Different Forms of the Equation of a Straight Line
(i) Slope Intercept Form The equation of a line with slope m and
making an intercept c on Y-axis, is
y = mx + c
If the line passes through the origin, then its equation will be
y = mx
(ii) One Point Slope Form The equation of a line which passes
through the point ( x1 , y1 ) and has the slope m is given by
( y − y1 ) = m ( x − x1 )
(iii) Two Points Form The equation of a line passing through the
points ( x1 , y1 ) and ( x2 , y2 ) is given by
y − y1
( y − y1 ) = 2 ( x − x1 )
x2 − x1
This equation can also be determined by the determinant
method, that is
x y 1
x1 y1 1 = 0
x2 y2 1
A straight line is the locus of all those points which are collinear with
two given points.
General equation of a line is ax + by + c = 0
Note
l We can have one and only one line through a fixed point in a given direction.
l We can have infinitely many lines through a given point.
Slope (Gradient) of a Line
The trigonometrical tangent of the angle that a line makes with the
positive direction of the X-axis in anti-clockwise sense is called the
slope or gradient of the line.
So, slope of a line, m = tan θ
where, θ is the angle made by the line with positive direction of X-axis.
Important Results on Slope of Line
(i) Slope of a line parallel to X-axis, m = 0.
(ii) Slope of a line parallel to Y-axis, m = ∞.
(iii) Slope of a line equally inclined with axes is 1 or −1as it makes an angle of
45° or 135°, with X-axis.
(iv) Slope of a line passing through ( x1 , y1) and ( x2 , y2 ) is given by
y −y
m = tanθ = 2 1 .
x2 − x1
Angle between Two Lines
The angle θ between two lines having slopes m1 and m2, is
m − m1
tanθ = 2 .
1 + m1m2
, (i) Two lines are parallel, iff m1 = m2.
(ii) Two lines are perpendicular to each other, iff m1m2 = − 1.
Equation of a Straight Line
General equation of a straight line is Ax + By + C = 0.
(i) The equation of a line parallel to X-axis at a distance b from it, is
given by
y=b
(ii) The equation of a line parallel to Y-axis at a distance a from it, is
given by
x=a
(iii) Equation of X-axis is
y=0
(iv) Equation of Y-axis is
x=0
Different Forms of the Equation of a Straight Line
(i) Slope Intercept Form The equation of a line with slope m and
making an intercept c on Y-axis, is
y = mx + c
If the line passes through the origin, then its equation will be
y = mx
(ii) One Point Slope Form The equation of a line which passes
through the point ( x1 , y1 ) and has the slope m is given by
( y − y1 ) = m ( x − x1 )
(iii) Two Points Form The equation of a line passing through the
points ( x1 , y1 ) and ( x2 , y2 ) is given by
y − y1
( y − y1 ) = 2 ( x − x1 )
x2 − x1
This equation can also be determined by the determinant
method, that is
x y 1
x1 y1 1 = 0
x2 y2 1
