# Slope (Gradient) of a Line

The 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.

Notes:
• The slope of a line is generally denoted bym. Thus, m = tan Î¸.
• A line parallel to the x-axis makes an angle of 0Â° with the x-axis, therefore its slope is tan 0Â° = 0.
• A line parallel to the y-axis, i.e., perpendicular to the x-axis makes an angle of 90Â° with the x-axis, so its slope is tan Ï€/2 = âˆž.
• The slope of a line equally inclined with axes is 1 or â€“1 as it makes 45Â° or 135Â° angle with the x-axis.
• The angle of inclination of a line with the positive direction of the x-axis in anticlockwise sense always lies between 0Â° and 180Â°, i.e., 0 â‰¤ Î¸ < Ï€ Î¸ â‰  Ï€/2.

# Slope of a line in terms of coordinates of any two points on it

Let P(x1, y1) and Q(x2, y2) be two points on a line making an angle Î¸ with the positive direction of the x-axis, then its slope is

# Slope of a line when its equation is given

Slope of a line whose equation is given by ax + by + c = 0 is

# Angle between two lines

The angle Î¸ between the lines having slope m1 and m2 is given by

The acute angle between the lines is given by tan Î¸ = .

# Condition of parallelism of lines

It two lines of slopes m1 and m2 are parallel, then the angle Î¸ between them is of 0Â°.

âˆ´ tan Î¸ = tan 0Â° = 0

â‡’

â‡’ m2 = m1

Thus when two lines are parallel, their slopes are equal.

# Condition of perpendicularity of two lines

If two lines of slopes m1 and m2 are perpendicular, then the angle Î¸ between them is of 90Â°
âˆ´ cot Î¸ = 0 â‡’ = 0 â‡’ m1 m2 = â€“1

Thus when lines are perpendicular, the product of their slope is â€“1. If m is the slope of a line, then slope of a line perpendicular to it is â€“ (1/m).

# Locus and equation to a locus

Locus The curve described by a point which moves under given condition or conditions is called its locus.

Equation to locus of a point The equation to the locus of a point is the relation which is satisfied by the coordinates of every point on the locus of the point.

How to find locus

Step I: Assume the coordinates of the point say, (h, k) whose locus is to be found.

Step II: Write the given condition in mathematical form involving h, k.

step III: Eliminate the variable(s), if any.

Step IV: Replace h by x and k by y in the result obtained in step III.

The equation so obtained is the locus of the point which moves under some stated condition(s).

# Shifting of origin

If (x, y) are coordinates of a point referred to old axes and (X, Y) are the coordinates of the same point referred to new axes, then x = X + h and y = Y + k.

If therefore the origin is shifted at a point (h, k) we must substitute X + h and Y + k for x and y respectively.

The transformation formula from new axes to old axes is X = x â€“ h, Y = y â€“ k. The coordinates of the old origin referred to the new axes are (â€“h, â€“k).