Various forms of equation of straight lines

A straight line is a curve such that every point on the line segment joining any two points on it lies on it.

Slope (Gradient) of a line:

The trignometrical tangent of the angle that a line makes with the positive direction of the x - axis in anticlockwise sense is called the slope or gradient of the line.

The slope of a line is generally denoted by m.

m = tanθ

The angle of inclination of a line with the positive direction of x - axis in anticlockwise sense always lies between 0⁰ and180⁰.

The slope of a line joining (x₁, y₁) and (x₂, y₂) is given by 

m = y₂ - y₁/ x₂ - x₁ = Difference of Ordinates/ Difference of abscissae.

Various forms of equation of straight lines:

Slope intercept form of a line:

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 0 = m0 + c => c = 0

Therefore, the equation of a line passing through the origin is y = mx

If the line is parallel to x – axis, then m = 0 therefore the equation of a line parallel to x - axis is y = c.

Reduction of general form to slope intercept form:

The general form of equation of a line is Ax + By + C = 0

⇒ By = - Ax – C

⇒ y = (- A/B) x + (- C/D)

This is of the form y = mx + c, where m = - A/B and C = - C/B.

Point – Slope form of a line:

The equation of a line which passes through the point (x₁, y₁) and has the slope ‘m’ is (y – y₁)= m (x – x₁)

Example:

Find the equation of a line passing through (2, -3) and inclined at an angle 135⁰ with the positive direction of x – axis.

Solution:

Here, m = slope of the line

= tan 135⁰ = tan (90⁰ + 45⁰)

= - cot 45⁰

= -1

And x₁ = 2, y₁ = -3.

So, the equation of the line is y – y₁ = m (x – x₁)

Or, y – (-3) = -1 (x - 2)

Or, y + 3 = - x + 2 or x + y + 1 = 0

Two point form of a line:

The equation of a line passing through two points (x₁, y₁) and (x₂, y₂) is (y - y₁) [y₂ - y₁/ x₂ - x₁] (x - x₁).

The equation of a line passing through two points (x₁, y₁) and (x₂, y₂) can also be written in the determinant form 

Intercept form of a line:

The equation of a line which cuts off intercepts a and b respectively from the x and y - axis is x/a + y/b = 1.

Reduction of general equation of a line to intercept form:

The general equation of a line is Ax + B y + C = 0

⇒ Ax + By = - C

⇒ (Ax /-C) + (By/ -C) = 1

⇒ x/ (-C/A) + y/ (-C/B) = 1

This is of the form x/a + y/b = 1

Intercept on x-axis = -C/A

Intercept on y-axis = -C/B

The intercept made by a line on x-axis can also be obtained by putting y = 0 in its equation.

Similarly, y-intercept is the value of y obtained from the line when x is replaced by zero.