We call it *angular coefficient *straight **r **the real number **m** such that:

The angle is counterclockwise oriented and obtained from the positive half axis **Ox** to the straight **r**. So we always have . Like this:

for (tangent is positive in 1st quadrant)

for (tangent is negative in 2nd quadrant)

Examples:

## Determination of the angular coefficient

Let's consider three cases:

a) the angle is known

b) the coordinates of two distinct points of the line are known: **THE**(x_{THE}, y_{THE}) and **B**(x_{B}, y_{B})

How (corresponding angles) we have to .

But, m = tg So:

Thus, the angular coefficient of the line passing, for example, through **THE**(2,3) and **B**(-2, 5) é:

c) the general equation of the line is known

If a line goes through two distinct points **THE**(X_{THE}, Y_{THE}) and **B**(X_{B}, Y_{B}), we have:

Applying Laplace's theorem in the first line, comes:

(Y_{THE} - Y_{B}) x + (X_{B} - X_{THE}) y + X_{THE}Y_{THE} - X_{B}Y_{B} = 0

From the general equation of the line we have:

Overriding these values in we have:

## Equation of a line r, known the angular coefficient and a point of r

Be** r** an angular coefficient line **m**. Being **P**(X_{0}, Y_{0}), P r and **Q**(x, y) any point of r (QQ), we can write:

As an example, let's determine the general equation of the line **r** that goes through **P**(1, 2), where m = 3. So we have X_{0}= 1 and Y_{0}= 2. Soon:

y-y_{0}= m (x-x_{0})

y-2 = 3 (x - 1)

y-2 = 3x - 3

3x - y - 1 = 0

which is the general equation of** r**.