Slope of a line

The measure of steepness and direction of a straight line is given by its slope. Slope is usually represented by the letter m. In the given figure, if the angle of inclination of the given line with the x-axis is θ then, the slope of the line is given by tan θ.

The slope of a line is given as m = tan θ. If two points A \((x_1,y_1)\)

m = tan θ = \( \frac{y_2~-~y_1}{x_2~-~x_1}\)

Where θ is the angle which the line AB makes with the positive direction of the x-axis. θ lies between 0° and 180°

It must be noted that θ = 90° is only possible when the line is parallel to y-axis i.e. at \( x_1 \)

Conditions for perpendicularity, parallelism and collinearity of straight lines

For parallel lines:

Consider two parallel lines given by \( l_1\)

Therefore, if the slopes of two lines on the Cartesian plane are equal then the lines are parallel to each other.

Thus, if two lines are parallel then, \( m_1\)

Generalizing this,n lines then they are parallel only when the slopes of all the lines are equal.

If the equation of the two lines are given as ax + by + c = 0 and a’ x + b’ y + c’=0, then they are parallel when ab’ = a’b. (How? You can arrive at this result if you find the slopes of each line and equate them.)

For perpendicular lines:

In the figure, we have two lines \( l_1\)

Their slopes can be given as:

m1 = tan(α+90°) and \( m_2 \)

⇒\( m_1 \)

⇒\( m_1 \)

⇒\( m_1 ~×~m_2 \)

Thus, for two lines to be perpendicular the product of their slope must be equal to -1.

If the equation of the two lines is given by ax + by + c = 0 and a’ x + b’ y + c’ = 0, then they are perpendicular if, aa’+ bb’ = 0. (Again, you can arrive at this result if you find the slopes of each line and equate their product to -1.)

For collinearity:

For two lines AB and BC to be collinear the slope of both the lines must be equal and there should be at least one common point through which they should pass. Thus for three points A,B and C to be collinear the slopes of AB and BC must be equal.

If the equation of the two lines is given by ax + by + c = 0 and a’ x+b’ y+c’ = 0, then they are collinear when ab’ c’ = a’ b’ c = a’c’b.

Angle between two lines:

When two lines intersect at a point then the angle between them can be expressed in terms of their slopes and is given by the following formula:

tan θ = |\( \frac{ m_2~-~m_1}{1~+~m_1~ m_2}\)

If \( \frac{ m_2~-~m_1}{1~+~m_1~ m_2}\)

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