Table of Contents

## Slope of a Line:

The inclination of a line to the X-axis ranges from 0**°** to 180**°**. By the term inclination, we refer to the angle that the line makes with the positive direction of the X-axis.

The slope or the gradient of a line is the tangent of the angle that the line makes with the positive direction of the X-axis. Thus, if θ is the inclination of the line, then its slope is given by m = tan θ as shown above.

If θ = 0, then m = 0 If 0 < θ < π/2, then m > 0If π/2 < θ < π, then m < 0If θ = π/2, then 1/m = cot θ = 0 |

If a non-vertical line passes through two distinct points (x_{1}, y_{1}) and (x_{2}, y_{2}), then the slope of the line is given by, m = (y_{2} – y_{1})/(x_{2} – x_{1}) = (y_{1} – y_{2})/(x_{1} – x_{2}).

In figure (b), when 0 < θ < π/2slope m = tan θ ⇒ slope m = QN/PN = (y _{2} – y_{1})/(x_{2} – x_{1})⇒ slope m = (y _{1} – y_{2})/(x_{1} – x_{2}) |

In figure (c), when π/2 < θ < πslope m = tan θ ⇒ slope m = – tan ( π – θ)⇒ slope m = – PN/NQ ⇒ slope m = – (y _{1} – y_{2})/(x_{2} – x_{1}) = (y_{2} – y_{1})/(x_{2} – x_{1})⇒ slope m = (y _{1} – y_{2})/(x_{1} – x_{2}) |

(i) If θ = 0, then slope m = (yNote:_{2} – y_{1})/(x_{2} – x_{1}) = 0/PQ = 0 as shown in figure (d).(ii) If θ = π/2, then tan θ is undefiend. In such a case, when the line is vertical, for the sake of mathematical calculation, we take 1/m = 0. |

## Parallel and Perpendicular Lines:

(1) *Parallel Lines-**Two parallel lines have the same slope. *

We consider two parallel lines *l*_{1} and *l*_{2} having slopes m_{1} and m_{2} (none of them being parallel to the Y-axis) respectively and let the lines make angles α_{1} and α_{2} respectively with the positive direction of the X-axis. Thus, m_{1} = tan α_{1} and m_{2} = tan α_{2}.

Since the lines are parallel to each other, α _{1} = α_{2}∴ tan α _{1} = tan α_{2}i.e. m _{1} = m_{2}Thus, two parallel lines will always have the same slope. |

*Two lines having the same slope are parallel to each other.*

We now consider two lines *l*_{1} and *l*_{2} having the same slope.

∴ m_{1} = m_{2}, i.e., tan α_{1} = tan α_{2} ⇒ α_{1} = α_{2}

Thus, the lines *l*_{1} and *l*_{2} are parallel to each other.

** Conclusion:** If two lines are parallel then they will have the same slope and conversely, if two lines have the same slope then they will be parallel to each other.

(2) *Perpendicular Lines-**The product of the slope of two perpendicular lines is -1.*

We consider two perpendicular lines *l*_{1} and *l*_{2} having slopes m_{1} and m_{2}. Let α_{1} and α_{2} respectively be the angles made by them with the positive direction of the X-axis.

Thus, either α_{1} = 90^{°} + α_{2} or, α_{2} = 90^{°} + α_{1}

αCase 1:_{1} = 90^{°} + α_{2}∴ tan α _{1} = tan (90^{°} + α_{2}) or, tan α_{1} = – cot α_{2} or, tan α_{1} = – 1/tan α_{2}or, m _{1} = -1/m_{2} or, m_{1}m_{2} = -1 αCase 2:_{2} = 90^{°} + α_{1}∴ tan α _{2} = tan (90^{°} + α_{1}) or, tan α_{2} = – cot α_{1} or, tan α_{2} = – 1/tan α_{1}or, m _{2} = -1/m_{1} or, m_{1}m_{2} = -1Thus, the product of the slopes of two perpendicular lines is -1. |

*If the product of the slopes of two lines is -1, then they are perpendicular to each other.*

Let m_{1}m_{2} = -1∴ m _{1} = -1/m_{2} or, tan α_{1} = -1/tan α_{2} = -cot α_{2} = tan (90^{°} + α_{2})∴ Either α _{1} = 90^{°} + α_{2} or, 180^{°} + α_{1} = 90^{°} + α_{2}i.e., Either α _{1} – α_{2} = 90^{°} or, α_{2} – α_{1} = 90^{°}In both cases, the lines are perpendicular to each other. |

** Conclusion:** If two lines are perpendicular, then the product of their slopes is -1 and conversely if the product of the slope of two lines is -1, then they are perpendicular to each other.

Thus, if the slope of a line is m_{1} (≠ 0), then the slope of any line perpendicular to it will be -1/m_{1}.

** Observation:** If a line has a slope of 2/3, then any line parallel to it will have a slope of 2/3, and any line perpendicular to it will have a slope of -3/2 [∵ 2/3 x (-3/2) = -1].

## Intercepts of a line on the coordinate axes:

If a line *l *intersects the X-axis at A and the Y-axis at B, then for A, y = 0, and for the point B, we must have x = 0. If a and b are the intercepts of the line on the X and Y-axes then the coordinates of A and B are (a, 0) and (0, b) respectively.

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