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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.
 
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Notes:
  • The slope of a line is generally denoted bym. Thus, m = tan θ.
  • A line parallel to the x-axis makes an angle of 0° with the x-axis, therefore its slope is tan 0° = 0.
  • A line parallel to the y-axis, i.e., perpendicular to the x-axis makes an angle of 90° with the x-axis, so its slope is tan π/2 = ∞.
  • The slope of a line equally inclined with axes is 1 or –1 as it makes 45° or 135° 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 0° and 180°, i.e., 0 ≤ θ < π θ ≠ π/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 θ with the positive direction of the x-axis, then its slope is
 
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Slope of a line when its equation is given

Slope of a line whose equation is given by ax + by + c = 0 is
 
69882.png

Angle between two lines

The angle θ between the lines having slope m1 and m2 is given by
69876.png
 
The acute angle between the lines is given by tan θ = 69870.png.
 
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Condition of parallelism of lines

It two lines of slopes m1 and m2 are parallel, then the angle θ between them is of 0°.
 
∴ tan θ = tan 0° = 0
 
69857.png69851.png
 
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 θ between them is of 90°
∴ cot θ = 0 69845.png = 0 m1 m2 = –1
 
Thus when lines are perpendicular, the product of their slope is –1. If m is the slope of a line, then slope of a line perpendicular to it is – (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 = xh, Y = yk. The coordinates of the old origin referred to the new axes are (–h, –k).




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