Slope of a Line: Theory

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The inclination of a Line:

  • A line in a coordinate plane forms two angles with the x-axis, which are supplementary.  The angle (say) θ made by the line l with the positive direction of x-axis and measured anti-clockwise is called the inclination of the line. Thus
    ≤ θ ≤ 180°.

Inclination

  • The lines parallel to the x-axis, or the coinciding with the x-axis, have the inclination of 0°. The inclination of a vertical line (parallel to or coinciding with y-axis) is 90°.

The slope of a Line:

  • If θ is the inclination of a line l, then tan θ, (θ ≠ 90°) is called the slope or gradient of the line l.
  • Note that the slope of a line whose inclination is 90° is not defined.
  • The slope of a line is denoted by letter, ‘m’. Thus by definition m = tan θ, θ ≠ 90°
  • The slope of the x-axis is zero and the slope of the y-axis is not defined

The slope of a line when coordinates of any two points on the line are given:

  • Let P(x1, y1) and Q(x2, y2) be two points on non-vertical line l whose inclination is θ. As the line is vertical, x1 x2The inclination of the line l may be acute or obtuse. Let us consider both of these cases.
  • Case – 1: When the inclination is acute


Draw perpendicular QR to the 
x-axis and PM perpendicular to RQ as shownMQ = y2 – y1 and MP = x2 – x1MPQ = θ. … (1)
Therefore, the slope of line 
l = m = tan θ.
But in 
MPQ, we have ∴ Slope of line l = (y2 – y1)/(x2 – x1)



  • Case – 2: When the inclination is obtuse


Draw perpendicular QR to the 
x-axis and PM perpendicular to RQ as shownMQ = y2 – y1 and MP = x1 – x2MPQ =  π – θ
Therefore, the slope of line 
l = m = tan θ = – tan (π – θ).
But in 
MPQ, we have ∴ Slope of line l = (y2 – y1)/(x2 – x1)∴ Slope of line l = (y2 – y1)/(x2 – x1)Thus, in either case, the slope of the line is m = (y2 – y1)/(x2 – x1)Sign of Slope:

  • The slope can be positive, zero or negative
  • Positive slope: Means the angle of inclination θ is such that < θ < 90°. i.e. the angle of inclination is acute.
  • Zero slope: Means the line is parallel to the x-axis.
  • Negative slope: Means the angle of inclination θ is such that 9< θ < 180°. i.e. the angle of inclination is obtuse.

Conditions for parallelism of lines in terms of their slopes

  • In a coordinate plane, suppose that non-vertical lines l1 and l2 have slopes mand m2, respectively. Let their inclinations be α and β, respectively.

  • If the line l1 is parallel to l2 (Fig 10.4), then their inclinations are equal, i.e., α = β, and hence, tan α = tan β

Therefore m1 = m2, i.e., their slopes are equal.
Conversely, if the slope of the two lines 
l1 and lare the same, i.e., m1 = m2.
By the property of tangent function (between 0° and 180°), α = β. Therefore, the lines are parallel.



  • Hence, two non-vertical lines l1 and l2 are parallel if and only if their slopes are equal

Conditions for perpendicularity  of lines in terms of their slopes:

  • If the lines l1 and l2 are perpendicular,  then by exterior angle property, β = α + 90°.

∴  tan β = tan (α + 90°)
∴  tan β = – cot
α =
∴  tan β = –  1/ 
∴  tanα . tan β = –  1
∴  m1 . m2 = – 1
Conversely, if 
m1 m2 = – 1, i.e., tan α tan β = – 1.
Then tan
α = – cot β = tan (β + 90°) or tan (β – 90°)
Therefore,
α and β differ by 90°.
Thus, lines 
l1 and l2 are perpendicular to each other.

  • Hence, two non-vertical lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other, or the product of their slopes is – 1.
Science > Mathematics > Straight LinesYou are Here
Physics Chemistry  Biology  Mathematics

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