Straight
Lines
Distance
Formula
The distance between two
points A(x1, y1) and B (x2,
y2) is given by
The distance of a point
A(x, y) from the origin 0 (0, 0) is given by OA = 
Section
Formula
The coordinates of the
point which divides the joint of (x1, y1) and (x2,
y2) in the ratio m : n internally, is
Mid-point of the joint
of (x1, y1) and (x2, y2) is
X-axis divides the line
segment joining (x1, y1) and (x2, y2)
in the ratio -y1 : y2.
Y-axis divides the line
segment joining (x1, y1) and (x2, y2)
in the ratio -x1 : x2.
The coordinates of the
centroid of the triangle whose vertices are (x1, y1), (x2,
y2) and (x3, y3) is
Area of
Triangle
The area of the
triangle, the coordinates of whose vertices are (x1, y1),
(x2, y2)and (x3, y3)
is the absolute value of
If the points (x1,
y1), (x2, y2) and (x3, y3)
are collinear, then x1 (y2 – y3) + x2 (y3 –
y1) + x3 (y1 – y2) = 0.
Shifting
of Origin
Let the origin is
shifted to a point O'(h, k). If P(x, y) are
coordinates of a point referred to old axes and P'(X, Y) are the coordinates of
the same points referred to new axes, then x = X + h, y = Y + k.
Straight
Line
Any curve is said to be
a straight line if two points are taken on the curve such that every point on
the line segment joining any two points on it lies on the curve. General
equation of a line is ax + by + c = 0.
Slope or
Gradient of Line
The inclination of angle
θ to a line with a positive direction of X-axis in the anti-clockwise
direction, the tangent of angle θ is said to be slope or gradient of the
line and is denoted by m.
i.e. m = tan θ
The slope of a line passing through points P(x1, y1) and
Q(x2, y2) is given by
Note: Slope of a line
parallel to X-axis is zero and slope of a line parallel to Y-axis is not
defined.
Angle
between Two Lines
The angle θ between
two lines having slope m1 and m2 is
·
If two lines are parallel, their slopes are equal
i.e. m1 = m2.
·
If two lines are perpendicular to each other, then
their product of slopes is -1 i.e. m1m2 = -1.
Various
Forms of the Equation of a Line
If a line is at a
distance k and parallel to X-axis, then the equation of the line is y = ± k.
If a line is parallel to
Y-axis at a distance c from Y-axis, then its equation is x = ± c.
Slope-intercept form: The equation of line
with slope m and making an intercept c on the y-axis, is y = mx + c.
One point-slope form: The equation of a line
which passes through the point (x1, y1) and has the slope
of m is given by y – y1 = m (x – x1).
Two points form: The equation of a line passing through the points
(x1, y1) and (x2, y2) is given by
The Intercept form: The
equation of a line which cuts off intercepts a and b respectively on the x and
y-axes is given by 
The normal form: The
equation of a straight line upon which the length of the perpendicular from the
origin is p and angle made by this perpendicular to the x-axis is α, is
given by x cos α + y sin α = p.
General Equation of a
Line
Any equation of the form Ax + By
+ C = 0, where A and B are simultaneously not zero is called the general
equation of a line.
Different
Forms of Ax + By + C = 0
Slope intercept form: If
B ≠ 0, then Ax + By +
C = 0 can be written as
If B = 0, then x = – C /
A which is a vertical line, whose slope is not defined and x-intercept is –
C/A.
Intercept form If C
≠ 0, then Ax + By + C
= 0 can be written as
where a = – C / A and b = –
C/B
If C = 0, then Ax + By + C = 0 can be written as Ax + By = 0 which is a line passing through origin and
therefore has zero intercept on the axes.
Normal form: The normal form of equation Ax
+ By + C = 0 is x cos α + y sin α = p where
Note: Proper choice of
signs to be made so that p should be always positive.
Position
of Points is Relative to a Given Line
Ø Let the equation of the
given line be ax + by + c = 0 and let the coordinates
of the two given points be P(x1, y1) and Q(x2,
y2).
Ø The two points are on
the same side of the straight line ax + by + c = 0,
If ax1 + by1 + c and ax2 + by2 +
c have the same sign.
Ø The two points are on
the opposite sides of the straight line ax + by + c =
0, If ax1 + by1 + c and ax2 +
by2 + c have opposite sign.
Condition
of concurrency for three given lines
a1x + b1y
+ c1 = 0, a2x + b2y + c2 =
0 and a3x + b3y+ c3 = 0 is a3(b1c2 – b2c1)
+ b3(a2c1 – a1c2) +
c3(a1b2 – a2b1) = 0
Point of
intersection of two lines
Let equation of lines be
ax1 + by1 + c1 = 0 and a2x
+ b2y + c2 = 0, then their point of intersection is
Distance
of a Point from a Line
The perpendicular
distanced of a point P(x1, y1)from
the line Ax + By + C = 0 is given by
Distance between
Two Parallel Lines
The distance d between
two parallel lines y = mx + c1 and y = mx + c2 is
given by
