
Distance between the points
is

The coordinates of a point dividing the line segment joining the points
internally, in the ratio m: n are

If , the coordinates will be.

Area of the triangle whose vertices are is

If the area of the triangle ABC is zero, then three points A, B and C lie on a line, i.e., they are collinear.

If is the inclination of a line l, then is called the slope or gradient of the line l. The slope of a line whose inclination is 90° is not defined. The slope of a line is denoted by m. Thus,

Slope

If the line is parallel to , then their inclinations are equal.

If the lines _{ }and are perpendicular such that _{ }makes an angle and makes an angle with the xaxis, then

Two nonvertical lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other

An acute angle (say θ) between lines L1 and L2 with slopes
is given by

Two lines are parallel if and only if their slopes are equal.

Two lines are perpendicular if and only if product of their slopes is –1.

Three points A, B and C are collinear, if and only if slope of AB = slope of BC.

Equation of the horizontal line having distance a from the xaxis is either

Equation of the vertical line having distance b from the yaxis is either

The point (x, y) lies on the line with slope m and through the fixed point if and only if its coordinates satisfy the equation

Equation of the line passing through the points
is given by

The point (x, y) on the line with slope m and yintercept c lies on the line if and only if

If a line with slope m makes xintercept d. Then equation of the line is

Equation of a line making intercepts a and b on the xand yaxis, respectively, is

The equation of the line having normal distance from origin p and angle between normal and the positive xaxis is given by .

Any equation of the form Ax + By + C = 0, with A and B are not zero, simultaneously, is called the general linear equation or general equation of a line.

The perpendicular distance (d) of a line Ax + By+ C = 0 from a point
is given by

Distance between the parallel lines Ax + By + C1 = 0 and Ax + By + C2 = 0, is given by
Examples
Question
If the angle between two lines is π/4 and slope of one of the lines is 1/2, find the slope of the other line.
Solution
θ = π/4
Substituting values,
Question
Three points P (h, k), Q (x1, y1) and R (x2, y2) lie on a line. Show that (h – x1) (y2 – y1) = (k – y1) (x2 – x1).
Solution
Since points P, Q and R are collinear, we have Slope of PQ = Slope of QR
Question
Find the equation of the line through (– 2, 3) with slope – 4.
Solution
Here m = – 4 and given point is (– 2, 3). By slopeintercept form formula, equation of the given line is
y – 3 = – 4 (x + 2) or 4x + y + 5 = 0, which is the required equation.
Question
Equation of a line is 3x – 4y + 10 = 0. Find its (i) slope, (ii) x – and yintercepts.
Solution

Given equation 3x – 4y + 10 = 0 can be written as
Comparing with y = mx + c, we have slope of the given line as m = 4/3

Given equation can be written as
y intercept is 5/2.
Question
Find the equation of a line perpendicular to the line x − 2y + 3 = 0 and passing through the point (1, – 2).
Solution
Given line x − 2 y + 3 = 0 can be written as
y = x/2 + 3/2
Slope of the line (1) is m1 = 2. Therefore, slope of the line perpendicular to line (1) is m2 = 1/m1 = 1/2
Equation of the line with slope – 2 and passing through the point (1, – 2) is y − (− 2) = −2(x −1) or y= − 2x.