The slope of a line
(i) Slope (m) = tan θ
(ii) Slope (m) = y_2-y_1\over x_2-x_1
Important result
- The slope of a line parallel to the x-axis, m=0
- The slope of a line parallel to the y-axis, m=∞
Equation of a straight Line
(i) The general equation of a straight line is Ax + By + C = 0
(ii) The equation of a line parallel to the x-axis at a distance b from it, is given by y = b
(iii) The equation of a line parallel to the x-axis at a distance a from it, is given by x = a
(iv) Equation of the X-axis is y = 0
(v) Equation of the Y-axis is x = 0
Different Forms of the Straight Line
(i) Slope Intercept Form: The equation of a line with slope m and making an intercept c on the Y-axis, is
y = mx + c
(ii) One Point Slope Form: The equation of a line which passes through the point (x1, y1) and has the slope m is given by
y – y1 = m(x-x1) + c
(iii) Two Points Form: The equation of a line which passes through the point (x1, y1) and (x2, y2) is given by
(y-y1) = y_2-y_1\over x_2-x_1(x-x1)
(iv) Intercept Form: The equation of a line which cuts off intercept a and b respectively on the X and Y axis is given by
{x\over a}+{y\over b}=1
The general equation Ax+By+C=0 can be converted into intercept form, as
{x\over -C/A}+{y\over -C/B}=1
(v) Normal Form: The equation of a straight line upon which the length of the perpendicular from the origin is p and the angle made by this perpendicular to the X-axis is α, is given by
x cos α + y sin α = p
(vi) Distance and Parametric Form: The equation of a straight line passing through (x1, y1) and making an angle θ with the positive direction of X-axis, is
{x - x_1\over cos θ} = {y - y_1\over sin θ} = r
Condition of concurrency
Condition of concurrency for three given lines a1x + b1y+c1 = 0, a2x + b2y+c2 = 0 and a3x + b3y+c3 = 0 is
a3(b1c2 – b2c1)+b3(c1a2 – a1c2)+c3(a1b2 – a2b1)=0
or, \begin{vmatrix} a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2}\\ a_{3} & b_{3} & c_{3} \end{vmatrix}
Distance of a point from a line
The distance of a point from a line is the length of the perpendicular drawn from the point to the line. let Ax+By+C=0 be a line, whose perpendicular distance from the point P(x1, y1) is d. then,
d = |Ax_1+By_1+C|\over \sqrt{A^2+B^2}
Note: The distance of the origin from the line
d = |C|\over \sqrt{A^2+B^2}
Distance between two parallel Lines
The distance between two parallel lines
y = mx + c1
y = mx + c2
d = |c_1-c_2|\over \sqrt{1+m^2}
