# 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 by
*m*. 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*(*x*_{1},*y*_{1}) and*Q*(*x*_{2},*y*_{2}) 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*m*_{1}and*m*_{2}is given byThe acute angle between the lines is given by tan

*Î¸*= .# Condition of parallelism of lines

It two lines of slopes

*m*_{1}and*m*_{2}are parallel, then the angle*Î¸*between them is of 0Â°.âˆ´ tan

*Î¸*= tan 0Â° = 0â‡’

â‡’

*m*_{2}=*m*_{1}Thus when two lines are parallel, their slopes are equal.

# Condition of perpendicularity of two lines

If two lines of slopes

*m*_{1}and*m*_{2}are perpendicular, then the angle*Î¸*between them is of 90Â°âˆ´ cot

*Î¸*= 0 â‡’ = 0 â‡’*m*_{1}*m*_{2}= â€“1Thus 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*).