Line: Inclination of a Line:
► The angle of inclination or simply an inclination of a line is the angle θ which the part of the line above x-axis and measured in anticlockwise direction.
Remarks:
► The inclination of x-axis is 0°.
► The inclination of every line parallel to x-axis is 0°.
► The inclination of y-axis is 90°.
► The inclination of every line parallel to y-axis is 90°.
Slope or Gradient of a Line:
► If θ is the inclination of a line, then the value of tan θ is called the slope or gradient of that line and it is denoted by m.
Theorem 1:
The slope of al line passing through two given points P(x1,y1) and Q(x2,y2) is given by m = (y2-y1)/(x2-x1).
Equation of a Line:
Intercepts Made By a Line on the Axes:
(x/a) + (y/b) = 1
x - intercept = OA = a
y - intercept = OB = b
Equation of a Straight Line in Various Forms:
Equation of x-axis is y = 0.
Equation of y-axis is x = 0.
Equation of a line parallel to y-axis, and at a distance ‘a’ from it, is x = a.
Equation of a line parallel to x-axis, at a distance ‘b’ from x axis is y = b.
Equation of a Line in Various Forms:
(i) Slope-intercept Form:
► The equation of a line with slope m and y-intercept c is given by, y = mx + c.
(ii) Point-Slope Form:
► The equation of a line with slope m and passing through a point A(x1,y1) is given by, (y-y1) = m(x-x1)
(iii) Two-Point Form:
► The equation of a line passing through two point A(x1,y1) and B(x2,y2) is given by (y-y1)/(x-x1) = (y2-y1)/(x2-x1)
Conditions for Parallelism and Perpendicularity:
► Two non-vertical lines with slopes m1 and m2 are perpendicular to each other, if and only if m1m2 = -1
► Two non-vertical lines are parallel if and only if their slopes are equal or m1 = m2.
