Straight Lines

Slope of a Line and Angle between Two Lines

Tips :

  • If the inclination of a non-vertical line is ‘θ’ then tan θ is called slope of the line and is usually denoted by m, thus m = tan θ
  • Slope of a horizontal line is ‘0’ [\because θ = 0°]
  • Slope of a vertical line is not defined [\because θ = 90°]
  • Slope of the line joining two points A(x1, y1), B (x2 y2) is \tt m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}
  • Two non-vertical lines are parallel if their slopes are equal.
  • Two non-vertical lines are perpendicular if their product of slopes is ‘−1’
  • If ‘3’ points are collinear (A, B, C) then slope of AB = Slope of BC = Slope of AC (or) Area ΔABC = 0 (or) one of the points lies on the line joining of the other two points (or) sum of two of the distances AB, BC, CA, is equal to the third
  • If ‘θ’ is an acute angle between the lines having slopes m1 and m2 then \tt \tan \theta=\begin{vmatrix}\frac{m_{1}-m_{2}}{1+m_{1}m_{2}} \end{vmatrix}
  • If ‘θ’ is an acute angle between the lines a1x + b1y + c1 = 0 a2 x + b2 y + c2 = 0 then \tt cos \theta=\begin{vmatrix}\frac{a_{1}a_{2}+b_{1}b_{2}}{\sqrt{a^{2}_{1} + b^{2}}_{1} .\sqrt{{a_2^2}+b_2^2}}\end{vmatrix}and \ \tan\theta=\begin{vmatrix}\frac{a_{1}b_{2}-a_{2}b_{1}}{a_{1}a_{2}+b_{1}b_{2}}\end{vmatrix} other angle between the lines is π – θ.
  • If a1 x + b1 y + c1 = 0 and a2 x + b2 y + c2 = 0 are parallel to y–axis then angle between them is ‘o’ or ‘π’.
  • If any one of the lines is parallel to y-axis and other makes on angle ‘θ’ with positive direction of x-axis then angle between lines is |90° – θ|.

Tricks :

  • If a line is equally inclined to the axes, then it will make an angle of 45° or 135° with x–axis (Positive direction of x-axis) and hence its slope will be tan 45° or tan 135° = ± 1.
  • Slope of the line ax + by + c = 0, b ≠ 0 is \tt \frac{-a}{b}

Part1: View the Topic in this video From 46:50 To 55:39

Part2: View the Topic in this video From 00:40 To 21:23

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Important Results on Slope of Line

1. Slope of a line passing through (x1 , y1) and (x2 , y2) is given by m=tan \theta= \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\cdot
2. Slope of a line parallel to Y-axis, m = ∞.
3. Slope of a line parallel to X-axis, m = 0.

Angle between Two Lines
The angle θ between two lines having slopes m1 and m2, is tan \theta= \begin{vmatrix}\frac{m_{2}-m_{1}}{1+m_{1}m_{2}} \end{vmatrix}\cdot
a). Two lines are parallel, iff m1 = m2.
b). Two lines are perpendicular to each other, iff m1m2 = −1.