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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

0° ≤ θ ≤ 180°.

- 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(x
_{1}, y_{1}) and Q(x_{2}, y_{2}) be two points on non-vertical line*l*whose inclination is θ. As the line is vertical, x_{1}≠ x_{2}, The 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 = y_{2} – y_{1} and MP = x_{2} – x_{1}∠MPQ = θ. … (1)

Therefore, the slope of line *l *= m = tan θ.

But in ∆MPQ, we have ∴ Slope of line *l* = (y_{2} – y_{1})/(x_{2} – x_{1})

**Case – 2:**When the inclination is obtuse

Draw perpendicular QR to the x-axis and PM perpendicular to RQ as shownMQ = y_{2} – y_{1} and MP = x_{1} – x_{2}∠MPQ = π – θ

Therefore, the slope of line *l *= m = tan θ = – tan (π – θ).

But in ∆MPQ, we have ∴ Slope of line *l* = (y_{2} – y_{1})/(x_{2} – x_{1})∴ Slope of line *l* = (y_{2} – y_{1})/(x_{2} – x_{1})Thus, in either case, the slope of the line is m = (y_{2} – y_{1})/(x_{2} – x_{1})Sign of Slope:

- The slope can be positive, zero or negative
- Positive slope: Means the angle of inclination θ is such that 0° < 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 90° < 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
*l*and_{1}*l*have slopes m_{2}_{1 }and m_{2}, respectively. Let their inclinations be α and β, respectively.

- If the line
*l*_{1}is parallel to*l*_{2}(Fig 10.4), then their inclinations are equal, i.e., α = β, and hence, tan α = tan β

Therefore m_{1} = m_{2}, i.e., their slopes are equal.

Conversely, if the slope of the two lines *l*_{1} and *l*_{2 }are the same, i.e., m_{1} = m_{2}.

By the property of tangent function (between 0° and 180°), α = β. Therefore, the lines are parallel.

- Hence, two non-vertical lines
*l*_{1}and*l*_{2}are parallel if and only if their slopes are equal

Conditions for perpendicularity of lines in terms of their slopes:

- If the lines
*l*_{1}and*l*_{2}are perpendicular, then by exterior angle property, β = α + 90°.

∴ tan β = tan (α + 90°)

∴ tan β = – cotα =

∴ tan β = – 1/

∴ tanα . tan β = – 1

∴ m_{1} . m_{2} = – 1

Conversely, if m_{1} m_{2} = – 1, i.e., tan α tan β = – 1.

Then tanα = – cot β = tan (β + 90°) or tan (β – 90°)

Therefore,α and β differ by 90°.

Thus, lines l_{1} and l_{2} 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 Lines > You are Here |

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